Relations and Functions PDF

**UNIT FOUR**\ **RELATIONS AND FUNCTION**\ **RELATIONS** Relation is a set whose elements are an ordered pairs.\ **Example: R= {(1, 5), (2.10)}**\ **Relation**is the way in which things are related to each other, The relating phrases are: - 'is smaller than', - 'is greater than', - 'is mu...

**UNIT FOUR**\ **RELATIONS AND FUNCTION**\ **RELATIONS** Relation is a set whose elements are an ordered pairs.\ **Example: R= {(1, 5), (2.10)}**\ **Relation**is the way in which things are related to each other, The relating phrases are: - 'is smaller than', - 'is greater than', - 'is multiple of', - 'is factor of', - 'is father of', - 'is son of', - 'is child of' and etc. +-----------------------------------------------------------------------+ | Definition: Let A and B be non-empty sets. A relation R from A to B | | is any subset of [**A** **×** **B**]{.math.inline}. | | | | In other words, R is a relation from A to B if and only if R ⊆ (A×B). | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | Domain and range of relations | | | | Definition: LetR be a relation from a set A to a set B. Then\ | | i. Domain of R = {x*:* (*x*, *y*) belongs to R for some | | *y*}[ **=** **{first** **elements}**]{.math.inline}\ | | ii.Range of R = {y*:* (*x*, *y*) belongs to R for some | | *x*}[ **=** **{second** **elements}**]{.math.inline} | +-----------------------------------------------------------------------+ Example: Let [*A* = {1, 2, 3, 4}]{.math.inline}and [*B* = {1, 3, 5}]{.math.inline}. [*R* = {(1, 3), (1, 5), (2, 3), (3, 5), (4, 5)}]{.math.inline}is a relation from A to B because [R is sub set of AxB]{.math.inline}. **Example: Let** [**A** **=** **{1,** **2,** **3}.** **R** **=** **{(1,** **3),** **(1,** **3),** **(2,** **3)}**]{.math.inline} **is a relation on A.** **Example:** Given the relation[*R* = {(1, 3), (2, 5), (7, 1), (4, 3)}]{.math.inline}, find the domain and range of the relation R.\ **Solution:** Since the domain contains the first coordinates, domain = {1, 2, 7, 4} and The range contains the second coordinates, range[= {3, 5, 1}]{.math.inline}. **Example**: Given [*A* = {1, 2, 4, 6, 7} ]{.math.inline}and [*B* = {5, 12, 7, 9, 8, 3}]{.math.inline} Find the domain and range of the relation [*R* = {(*x*, *y*): *x* ∈ *A*, *y* ∈ *B*, *x*  \> *y*}]{.math.inline} **Solution**: If we describe R by complete listing method, we will find[ *R*= {(4, 3), (6, 3), (7, 3), (6, 5), (7, 5)}]{.math.inline}. This shows that the domain of [*R* = {4, 6, 7}]{.math.inline} and the range of [*R* = {3, 5}]{.math.inline} **GRAPHS OF THE RELATION** The representation of a relation from set A to B by locating the ordered pairs in a coordinate system or by using arrows in a diagram displaying the members of both sets, or as a region on a coordinate system is called **graphs ofrelations**. **To sketch graphs of relations involving inequalities, do the following:**\ **1.** Draw the graph of a line(s) in the relation on the [xy]{.math.inline}-coordinate system.\ **2.** If the relating inequality is ≤ or≥ , use a solid line; if it is [\]{.math.inline}, use a broken line.\ **3.**Then take arbitrary ordered pairs represented by points, one from one side and the other from another side of the line(s), and determine which of the pairs satisfy the relation.\ **4.**The region that contains points representing the ordered pair satisfying the relation will be the graph of the relation.\ **Example**: Let A= {2, 3, 5} and B= {6, 7, 10}. The relation R from A to B be "x is a factor of y" then R= {(2, 6), (2, 10), (3, 6), (5, 10)} with: Domain [*x*= {2, 3, 5}]{.math.inline} and Range [*y*= {6, 10}]{.math.inline} **Example:** Sketch the graphs of the relation of\ **A.** [*R* = {(*x*, *y*): *y* \> *x*; *x* ∈ ℝandy ∈ ℝ}]{.math.inline}\ **B.** [*R* = {(*x*, *y*): *y* ≥ *x* + 1; *x* ∈ ℝandy ∈ ℝ}]{.math.inline}\ **Solution**: A 1\. Draw the graph of the line [*y* = *x*]{.math.inline}using broken line. 2\. Select two points one from one side and another from the other side. For example (1,4) and (3,-2) from the above and below the line [*y* = *x*]{.math.inline} 3\. The order pair (1, 4) the given relation. So the region where the point (1,4) is Contained the graphs of the relation In the inequality relation 𝑦\>𝑥⟹ 4 \> 1 True So we shade the region of (1, 4). The graphs of the relation is given as follows  **Solution**: B 1\. Draw the graph of the line [*y* = *x* + 1]{.math.inline} using solid line 2\. Select two points one from one side and another from the other side. For example (0, 5) and (2, 0) from the above and below the line y=x+1 3. The ordered pair [(0, 5) ]{.math.inline}satisfies the given relation. So the region where the point (0, 5) is Contained the graphs of the relation and we shade the region which contains (0, 5) -- -- **NOTE**: The domain and range of the relation is the set of real number. +-----------------------------------------------------------------------+ | To sketch the graph of a relation with two or more inequalities, | | | | i\. Using the same coordinate system, sketch the regions of each | | inequality.\ | | ii. Determine the intersection of the regions. | +-----------------------------------------------------------------------+ **Example**: sketch the graphs of the following relation and determine the domain and range. A. 𝑅 = {(𝑥, 𝑦): 𝑦 ≥ 𝑥 and 𝑦 ≥ −2𝑥 + 4} B. 𝑅 = {(𝑥, 𝑦): 𝑥 − 2𝑦 ≤ 0 and \|𝑥\| ≥ 1}\ **Solution**: A. **Solution**: -----------------------  ----------------------- **Note:** A graph of a relation when the relating phrase is an inequality is a region on the coordinate system. +-----------------------------------------------------------------------+ | FUNCTIONS | | | | Definition: A function is a relation such that no two ordered pairs | | have the same first-coordinates and different second-coordinates.\ | | ✓It is a relation in which no two ordered pairs have the same first | | element. | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | Domain and range of a function | | | | Domain: is the set of first element of the ordered pairs. | | | | Range: is the set of the second element of the ordered pairs | +-----------------------------------------------------------------------+ **Note**: **A relation is a function if and only if the domain is not repeated**. **Example**: For each of the following relation identify whether it is function or not. If it is function determine the domain and range. A. Consider the relation [*R* = {(1, 2), (7, 8), (4, 3), (7, 6)}]{.math.inline}\ **solution**: Since 7 is paired with both 8 and 6 the relation R is not a function. B. The relation [*R* = {(*x*, *y*): *y* *is* *the* *father* *of*  *x*}]{.math.inline}\ **Solution:** is a function because no child has more than one father. So the domain is child and range is father of the child. C. Consider the relation R = {(*x*, *y*): *y* is a grandmother of *x*}.\ **Solution**: This relation is not a function since everybody (*x*) has two grandmothers. D. Consider the following arrow diagrams. -- -- Which of these relations are functions? Solution: 𝑅1and 𝑅3 are a function where 𝑅2 is not a function, because 1 and 3 are paired with different the second elements. Then domain and range of the functions are given as Domain of R1= {1, 3, 5} and Domain R3= {1, 3, 5} and Range of R1= {𝑎, 𝑏} Range of R3= {𝑎, 𝑏, 𝑐} **Function notation** If *x* is an element in the domain of a function *f*, then the element in the range that is associated with *x* is denoted by *f*(*x*) and is called the image of *x* under the function *f*. This means *f* = {(*x, y*):*y*= *f* (*x*)} The notation *f* (*x*) is called function notation. Read *f* (*x*) as ''𝑓𝑜𝑓𝑥" **Note:** *f*, *g* and *h* are the most common letters used to designate a function. But, any letter of the alphabet can be used. **A function 𝑓 is a mapping of a relation from set A to set B we write: A→B** - If 𝑥∈𝐴 and 𝑦∈𝐵,then the function is denoted by 𝑦 = 𝑓(𝑥) - If (𝑥, 𝑦) ∈𝑓 means 𝑓 = {(𝑥, 𝑦): 𝑦 = 𝑓(𝑥)} - Domain of 𝑓 = {𝑥: 𝑥 is the set of all possible input} - Range of 𝑓 = {𝑦: 𝑦 is the set of all out puts} **Functional values** - If *f*: A → B is a function, then, for any 𝑥∈𝐴 the image of *x* under *f*,*f* (*x*) is called the functional value of *f* at *x.* - *f* (*x*) shows the 𝑦 −value for the given 𝑥 and shows that the value can be found when an 𝑥 −value is known\ For example, if [*f* (*x*) = *x* -- 3]{.math.inline}, then the functional value of *f* at [*x* = 5]{.math.inline} is [*f* (5) = 5--3 = 2]{.math.inline}. **Finding the functional value of *f* at *x* is also called evaluating the function at *x****, then when* [*x* = 5]{.math.inline}*,* [*y* = 2]{.math.inline}\ **Example1:** For each of the following find domain and range. **A.** [*f* (*x*)= 3]{.math.inline} **B**. [*f* (*x*)= 1 -- 3*x*]{.math.inline} **C**. [\$f\\ \\left( x \\right) = \\ \\sqrt{x\\ + \\ 4\\ }\$]{.math.inline} **D**. [\$f\\ \\left( x \\right) = \\ \\sqrt{x - 1}\$]{.math.inline} **E**. [\$f\\ (x)\\ = \\ \\frac{1}{2x}\$]{.math.inline} **Solution:** A. Domain = ℝ and Range = {3} B. Domain = ℝ and Range = ℝ C. Domain = {*x*: *x* ≥--4} and Range = {*y*: *y* ≥0} D. Domain = ℝ and Range = {*y*: *y* ≥ --1} E. Domain = ℝ\\{0} and Range = ℝ\\{0} **Example**: If *f* (*x*) = 2*x* +[\$\\sqrt{x\\ + \\ 4}\$]{.math.inline}, evaluate each of the following:\ **a.** [*f* ( − 4) ]{.math.inline} **b.** [*f* (5)]{.math.inline}\ **Solution** a. 𝑓(−4) = 2(−4) + [\$\\sqrt{- 4\\ + \\ 4\\ }\$]{.math.inline}= −8 b\. f[\$(5)\\ = \\ 2(5)\\ + \\ \\sqrt{4\\ + \\ 5\\ } = \\ 10\\ + \\ 3\\ = \\ 13\$]{.math.inline} +-----------------------------------------------------------------------+ | Combination of function | | | | Functions like numbers can be added, subtracted, multiplied and | | divided.\ | | Definition\ | | ❖ If 𝑓 and 𝑔 are a functions, then 𝑓 + 𝑔, 𝑓 − 𝑔, 𝑓. 𝑔and𝑓/𝑔 are the | | function defined by\ | | (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥) ←sum of functions\ | | ✓ (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥) ←difference of function\ | | ✓ (𝑓. 𝑔)(𝑥) = 𝑓(𝑥). g(𝑥) ←product of function\ | | ✓[\$\\mathbf{(}\\frac{\\mathbf{f}}{\\mathbf{g}}\\mathbf{)\\ | | (}\\mathbf{x}\\mathbf{)\\ | | =}\\frac{\\mathbf{f(x)}}{\\mathbf{g(x)}}\$]{.math.inline} , 𝑔(𝑥) ≠ 0 | | ← Quotients of function.\ | | Note: The domain of 𝑓 + 𝑔, 𝑓 − 𝑔,.𝑔 is the intersection of the domain | | of 𝑓 and 𝑔 and the domain of 𝑓/𝑔 is the intersection of the domain of | | 𝑓 and 𝑔 but f (𝑥) ≠ 0 | +-----------------------------------------------------------------------+ **GRAPHS OF FUNCTIONS** **Graphs of Linear Functions** **Linear function** **Definition**: If *a*and*b* are fixed real numbers, *a* ≠ 0, then *f*(*x*) = *ax* + *b* for *x* ∈ℝ is called a linear function. If *a* = 0, then *f*(*x*) = *b* is called a constant function. Sometimes linear functions are written as *y* = *ax* + *b*. where its domain is the set of real number. **Example:** In each of the following given linear function, determine i\. slope ii\. Whether increasing or decreasing iii\. 𝑥 and 𝑦 intercept iv\. Sketch the graphs a\. 𝑦 − 3𝑥 − 5 = −2 b\. 2𝑥 + 𝑦 = 6 c. (𝑥) − 6 = −2\ **Solution** a. [*y* − 3*x* − 5 =  − 2]{.math.inline} → [*y* = 3*x* + 3]{.math.inline} Then [*a* = 3]{.math.inline} and [*b* = 3]{.math.inline} a\. the slope is [*a* = 3]{.math.inline} and increasing because[*a* = 3 \> 0]{.math.inline}, b\. 𝑥 −intercept [\$( - \\frac{b}{a}\\ ,\\ 0) = \\ ( - 1,\\ 0)\$]{.math.inline} and c\. 𝑦 −intercept [(0, *b*) = (0, 3)]{.math.inline} d\. the graphs of the function is gives as above +-----------------------------------------------------------------------+ | Properties of linear functions | | | | - The graphs of *f*(*x*) = *ax* + *b* is straight line | | | | | | | | - 𝑎 is the slope of the line | | | | - If 𝑎\> 0 the graph of *f*(*x*) = *ax* + *b* is increasing | | | | - If 𝑎\< 0the graph of *f*(*x*) = *ax* + *b* is decreasing | | | | - If 𝑎 = 0the graph will be *f*(*x*) =*b* horizontal line, it is | | called constant linear function. | | | | - If 𝑥 = 0, then 𝑓(𝑥) = 𝑏 ← 𝑦 −intercept (0,b) where the graph | | crosses the y-axis | | | | - If f(x)= 0, then 𝑎𝑥 + 𝑏 = 0 → 𝑥 = − 𝑏\ | | 𝑎← 𝑥 −intercept which is (− 𝑏/a , 0) | +-----------------------------------------------------------------------+ **GRAPHS OF QUADRATIC FUNCTIONS** +-----------------------------------------------------------------------+ | Quadratic function | | | | Definition: A function defined by | | [**(x)** **=** **ax**^**2**^ **+** bx **+** **c**]{.math | |.inline}where 𝑎, 𝑏, 𝑐𝜖ℝ and 𝑎 ≠ 0 is called a quadratic function. | | | | - ais called the leading coefficient. | +-----------------------------------------------------------------------+ **Example** 1: [*f*(*x*) = 2*x*^2^ + 3*x* + 2 ]{.math.inline}is a quadratic function with 𝑎 = 2, 𝑏 = 3, 𝑐 = 2 -------------------------------------------------------------------------------------------------------------------------------------- Note: Any function that can be reduced to the form (𝑥) = 𝑎[**x**^**2**^]{.math.inline} + 𝑏𝑥 + 𝑐 is also called a quadratic function -------------------------------------------------------------------------------------------------------------------------------------- **Example** 2: f(𝑥) = (𝑥 − 2)(𝑥 + 2) is expressed as 𝑓(𝑥) = [*x*^2^]{.math.inline} − 4 with 𝑎 = 1, 𝑏 = 0, 𝑎𝑛𝑑 𝑐 = −4 **Sketching graphs of quadratic function using a table of values** **Example**3: Draw the graphs of each of the following a\. (𝑥) = [*x*^2^]{.math.inline} c. (𝑥) = 2[*x*^2^]{.math.inline} b\. (𝑥) = −[*x*^2^]{.math.inline}. d. (𝑥) = −2[*x*^2^]{.math.inline} **Solution**: A. Evaluating the function values in the interval of [− 2  ≤ x ≤ 2]{.math.inline} X -2 -1 0 1 2 --------------------------- ---- ---- --- --- --- Y=[*x*^2^]{.math.inline} 4 1 0 1 4 Then the graph is given as B. Evaluating the function values in the interval of − 2 ≤*x* ≤ 2 X -2 -1 0 1 2 --- ---- ---- --- ---- ---- Y 4 1 0 -1 -4 Similarly the graph of the function is given as **Example 4:** sketch the graphs of the following using table value. A. (𝑥) = 2[*x*^2^]{.math.inline} + 3 B. (𝑥) = 2[*x*^2^]{.math.inline} − 3 C. f(𝑥) = −2[*x*^2^]{.math.inline} + 3 D. (𝑥) = −2[*x*^2^]{.math.inline} − 3\ **a. Solution: the table values of** [**f(x)**]{.math.inline} **is** **X** **-2** **-1** **0** **1** **2** ------- -------- -------- ------- ------- -------- **Y** **11** **5** **3** **5** **11**  b\. the table value of the function [*f*(*x*) = 2*x*^2^ − 3]{.math.inline} X -2 -1 0 1 2 --- ---- ---- ---- ---- --- Y 5 -1 -3 -1 5 **In General from the above graphs we summarize the following** +-----------------------------------------------------------------------+ | Case 1: If *a* \> 0 | | | | 1. The graph opens upward. | | | | 2. The vertex is (0, 0) for (𝑥) = 𝑎𝑥2 and (0, *c)* for *f* (*x*) = | | 𝑎𝑥2 + *c* | | | | 3. The domain is all real numbers. | | | | 4. The range is {*y*: *y* ≥ 0} for *f* (*x*) = 𝑎𝑥2 and {*y*: *y* ≥ | | *c*} for *f* (*x*) = 𝑎𝑥2 + *c* | | | | 5. The vertical line that passes through the vertex is the axis of | | the parabola (or the axis of symmetry). | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | Case 2: If *a* \< 0 | | | | 1. The graph opens downward. | | | | 2. The vertex is (0, 0) for (𝑥) = 𝑎𝑥2 and (0, *c)* for *f* (*x*) = | | 𝑎𝑥2 + *c* | | | | 3. The domain is all real numbers. | | | | 4. The range is {*y*: *y* ≤ 0} for *f* (*x*) = 𝑎𝑥2 and {*y*: *y* ≤ | | *c*} for *f* (*x*) = 𝑎𝑥2 + *c* | | | | 5. The vertical line that passes through the vertex is the axis of | | the parabola (or the axis of symmetry). | +-----------------------------------------------------------------------+ To sketch the graphs of quadratic function in the above we have used tables of values to sketch graphs of quadratic functions. Now we shall see how to use the shifting rule to sketch the graphs of quadratic functions.  **Note**: 1\. The graph of [*f* (*x*) = (*x* + *k*)^2^ + *c*]{.math.inline} opens upward. 2 The vertex of the graph of f (x) = (x + k)^2^ + c is (--k, c) and the vertex of the graph of f (x) = (x + k)^2^-- c is (k, --c). Similarly the vertex of the graph of f (x) = (x + k)^2^-- c is (--k, --c) and the vertex of the graph of\ f (x) = (x + k)^2^+ c is (k, c).\ **C.** To construct the graphs of (𝑥) = (𝑥 − 2)^2^ + 13, by drawing the graphs of f(x) = x^2^, and shifting on the line of x-axis by 2 unit to the right direction and shifting 13 units on the axis of symmetry 𝑥 = 2 then the graphs is given as follows. ----------------------------------------------------------------------------------- \ -3 -2 -1 0 1 2 3 [*x*]{.math.display}\ ------------------------------------------------ ---- ---- ---- ---- ---- ---- ---- \ 9 4 1 0 1 4 9 [*f*(*x*) = *x*^2^]{.math.display}\ \ 25 16 9 4 1 0 1 [*f*(*x*) = (*x*− 2)^2^]{.math.display}\ \ 38 29 22 17 14 13 14 [*f*(*x*) = (*x*− 2)^2^ + 13]{.math.display}\ ----------------------------------------------------------------------------------- **D.** Similarly to draw the graphs of (𝑥) = (𝑥 + 1)^2^− 7 by shifting from the graphs of f(x) = x^2^ to the left direction by one unit we get the graphs of (𝑥) = (𝑥 + 1)^2^, lastly we shift 7 unit downward on the line of (x=-1) axis of symmetry. Then using table value ----------------------------------------------------------------------------------- \ -3 -2 -1 0 1 2 3 [*x*]{.math.display}\ ------------------------------------------------ ---- ---- ---- ---- ---- ---- ---- \ 9 4 1 0 1 4 9 [*f*(*x*) = *x*^2^]{.math.display}\ \ 4 1 0 1 4 9 16 [*f*(*x*) = (*x* + 1)^2^]{.math.display}\ \ -3 -6 -7 -6 -3 -3 9 [*f*(*x*) = (*x* + 1)^2^ − 7]{.math.display}\ ----------------------------------------------------------------------------------- +-----------------------------------------------------------------------+ | Generally: | | | | Quadratic function is a function of the form 𝑓(𝑥) = 𝑎𝑥^2^ + 𝑏𝑥 + 𝑐, | | 𝑤𝑖𝑡ℎ𝑎 ≠ 0 | | | | - The domain of the quadratic function is the set of all real | | number. | | | | - The graph of all quadratic function is called parabolas. | | | | - The lowest or the highest point of the parabola called the | | vertex. | | | | - The vertical line passing through the vertex in each parabola is | | called axis of symmetry. | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | - Vertex (ℎ, 𝑘) | | | | - Axis of symmetry, 𝑥 = ℎ | | | | - If 𝑎\> 0, the graph is open upward | | | | - If 𝑎\< 0, the graph is open downward | | | | - The graph is narrows as \|𝑎\| is increase | +-----------------------------------------------------------------------+ +-----------------------------------------------------------------------+ | To obtain the graph of (𝑥) = (𝑥 − ℎ)^2^+ 𝑘, shifting the graphs of | | 𝑓(𝑥) = 𝑎𝑥^2^ | | | | i. We shift h unit to the right if (𝑥) = (𝑥 − ℎ)^2^ + 𝑘 h unit to | | the left if 𝑓(𝑥) = 𝑎(𝑥 + ℎ)^2^ + 𝑘 | | | | ii. We shift 𝑘, unit up if 𝑘\> 0, and down if 𝑘\< 0 | +-----------------------------------------------------------------------+ **Minimum and maximum values of quadratic functions** When the graph of a quadratic function opens upward, the function has a minimum value, whereas if the graph opens downward, it has a maximum value. The minimum or the maximum value of a quadratic function is obtained at the vertex of its graph. **Example:** Determine the minimum or the maximum value and draw the graph of f (𝑥) = 𝑥^2^ + 7𝑥 − 10 **Solution:** 𝑓(𝑥) = 𝑥^2^ + 7𝑥 − 10 = (𝑥 + 7 /2)^2^ -- 89/ 4 Then when we sketch its graph by using shifting rule it is given as follows ,Then vertex (ℎ, 𝑘) = (− 7, -89/4) The minimum value is -89/4. **REVIEW EXERCISE** 1\. Which of the following relations is a **function**? 2\. What is the domain of a relation R given by[*R* = {(*x*, *y*): *y* ≥ *x*^2^ − 1 *andy* \ 2\\}\$]{.math.inline} B. [\$\\{(x,y):y \\leq \\frac{x}{2}\\text{\\ and\\ }\\left\| x \\right\| \\geq 2\\}\$]{.math.inline} C. [\$\\{(x,y):y \\leq \\frac{x}{2}\\ and\\ x \> 2\\}\$]{.math.inline} D. [\$\\{(x,y):y \\leq \\frac{x}{2}\\ and\\ x \< - 2\\}\$]{.math.inline} 30\. Which of the following is the equation of a line whose x-intercept and y-intercept are 1 and -2 respectively? A. [*x* − *y* + 2 = 0]{.math.inline} B. [ − 2*x* + *y* − 1 = 0]{.math.inline} C. [*x* − 2*y* − 1 = 0]{.math.inline} D. [2*x* − *y* − 2 = 0]{.math.inline} 31\. Which of the following statements is true about the function [*f*(*x*) = (*x* − 1)^2^ + 3]{.math.inline}? A. The x-intercept is 1 C. Its range is (3,[∞]{.math.inline}) B. It is increasing in the interval [\[1, ∞]{.math.inline}) D. Its graph is symmetric with respect to the y-axis 32\. Let[*R* = {(*x*, *y*): *y* \ 1}]{.math.inline}, [*Range*  = {*y*: *y* \> 0}]{.math.inline}, B. [*Domain* = {*x*: *x* ≥ 1]{.math.inline}}, [*Range*  = {*y*: *y* \> 0}  ]{.math.inline} C. [*Domain*  = {*x*: *x* ≥ 1}]{.math.inline}, [*Range*  = {*y*: *y* ≥ 0}]{.math.inline}, D. [*Domain* = {*x*: *x* \> 1}, *Range*  = {*y*: *y* \> 1}]{.math.inline} **56.** If 𝑓 is a function which is given by (𝑥) = 𝑥^2^ + (1 − 𝑥)^2^ − 13, then what is the minimum values of this function? A. ½ B. -12 C. -25/2 D. -21/2 57\. Let 𝐴 = {−1, 0, 1}. A relation 𝑅 on set 𝐴 is defined by [*R* = {(*x*, *y*)  : *y*\ 0]{.math.inline} B. [*a* \ 5 \\\\ 0,\\ if\\ 2 \< x \\leq 5 \\\\ x + 6,\\ if\\ x \\leq 2 \\\\ \\end{matrix} \\right.\\ \$]{.math.inline}.[ → *correct*]{.math.inline} B. [*f*(4) =  − *g*(4). → *correct*]{.math.inline} C. [*f*(*f*(3)) = 1]{.math.inline} D. [*f*(*g*( − 3)) =  − 6. → *incorrect*]{.math.inline} The correct answer is D. 70\. [\$\\left( x \\right) = x\^{2} - 3x = \\left( x - \\frac{3}{2} \\right)\^{2} - \\frac{9}{4}\$]{.math.inline} \ [*f*( − 2) = 10]{.math.display}\ V=[\$(\\frac{3}{2},\\ - \\frac{9}{4}\$]{.math.inline}) Y-intercept=(0,0) The correct answer is A.