General Mathematics Learning Modules PDF

**METROPOLITAN INSTITUTE OF ARTS AND SCIENCES** 4408, Capricorn St., Brgy. 177, Maria Luisa Subd. Camarin, Caloocan City **LEARNING MODULES** **NAME: LEARNING ACTIVITY \#01** **GRADE & STRAND: GENERAL MATHEMATICS** **SECTION: MR. KENT JOSHUA QUILICOL** **GENERAL MATHEMATICS** **A. WHAT IS FUN...

**METROPOLITAN INSTITUTE OF ARTS AND SCIENCES** 4408, Capricorn St., Brgy. 177, Maria Luisa Subd. Camarin, Caloocan City **LEARNING MODULES** **NAME: LEARNING ACTIVITY \#01** **GRADE & STRAND: GENERAL MATHEMATICS** **SECTION: MR. KENT JOSHUA QUILICOL** **GENERAL MATHEMATICS** **A. WHAT IS FUNCTION?** - is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component? The set X called the **domain** of the function. For each element of *x* in X, the corresponding element *y* in Y is called the **value** of the function at x, or the image of x. The set of all images of the elements of the domain is called the **range** of the function.  A. **x** 1 2 3 4 ------- --- --- --- --- **y** 3 4 5 6 B. **x** -2 -1 0 1 2 ------- ---- ---- --- --- --- **y** 7 3 1 5 5 C. **x** 3 3 7 9 ------- --- --- --- --- **y** 0 2 4 1 **Relation A and B are functions**, because for each x-values, there is only one y-value. Relation C is not a function, because the first two ordered pairs have the same x-value, but different in y-values. **Note:** "**One-to-many**" is *not allowed*, but "**Many-to-one**" is *allowed*. **B. VERTICAL LINE TEST FOR A FUNCTION** If a vertical line cuts the graph of a relation in more than one point, then the relation does not represent a function. The **Vertical Line Test** helps you to *determine if a given relation is a function.* For example: Applying the vertical line test is as easy as following these 3 easy steps: **Step One:** Observe the given graph and draw one or multiple vertical lines through the graph. **Step Two:** Note how many times the vertical line(s) that you drew intersected the graph (either coordinate points or a line/curve). **Step Three:** Based on the number of intersections, determine whether or not the relation is a function. - If your vertical line **intersects the graph only once**, then the relation is a **function**. - If your vertical line **intersects the graph more than once at any point**, the relation is **not** a function We already know that the graph in Figure 05 represents a function and that the graph in Figure 06 does **not** represent a function because we already applied the definition of a function to the corresponding mappings. **C. Basic Types of Function** +-----------------------------------+-----------------------------------+ | Constant Function | polynomial of 0th degree where | | | *f(x) = f(0) = a~0~ = c*. | | | Regardless of the input, the | | | output always results in a | | | constant value | +===================================+===================================+ | Linear Function | linear polynomial function is a | | | first-degree polynomial where the | | | input needs to be multiplied by m | | | and added to c. It can be | | | expressed by *f(x) = mx + c.* | +-----------------------------------+-----------------------------------+ | Quadratic Function | All functions in the form of *y = | | | ax^2^ + bx + c* where a, b, c ∈ | | | R, a ≠ 0 will be known as | | | Quadratic functions. | +-----------------------------------+-----------------------------------+ | Rational Function | These are the real functions of | | | the type | | | | | | \ | | | [\$\$f\\left( x | | | \\right)\\frac{g\\left( x | | | \\right)}{h\\left( x \\right)},\\ | | | h\\left( x \\right) \\neq | | | 0\$\$]{.math.display}\ | | | | | | where f (a) and g (a) are | | | polynomial functions of a defined | | | in a domain, where g(a) ≠ 0. | +-----------------------------------+-----------------------------------+ | Power Function | Is a function of the form *f(x) = | | | x^n^*^,^ where n is any real | | | number | +-----------------------------------+-----------------------------------+ | Absolute Function | Is a function of the form *f(x) = | | | \|x\|* | +-----------------------------------+-----------------------------------+ | Polynomial Function | Is any function f(x) of the form | | | *f(x) = Cn^Xn^ + Cn-1^Xn-1^ + | | |.......c1x + c0* | +-----------------------------------+-----------------------------------+ | Greatest Integer Function | Is a function of the form *f(x) = | | | \[x\].* The value of \[x\] is the | | | greatest integer that is less | | | than or equal to x. | +-----------------------------------+-----------------------------------+ Example: 1. [\$f\\left( x \\right) = \\ \\frac{x - 3}{x + 7}\$]{.math.inline} = Rational Function 2. *f(x) =* 3x^4^ + 8x^3^ -- 4x^2^ + 2 = Polynomial Function 3. *f(x)* = x^4^ = Power Function 4. *f(x)* = 18 = Constant Function 5. *f(x)* = 3x -- 11 = Linear Function 6. *f(x) =* \| 2x -- 11 \| = Absolute Value Function 7. *f(x) =* x^2^ + 5x + 6 = Quadratic Function 8. *f(x)* = \[2.7\] = Greatest Integer Function **D. PIECEWISE FUNCTION** Piecewise-defined function is a function which uses different rules on disjoint subsets of its domain. Example 1: Given the function [\$f\\left( x \\right) = \\left\\{ \\begin{matrix} 1\\ if\\ x \< 1 \\\\ 2\\ if\\ x = 1 \\\\ 3\\ if\\ x \> 1 \\\\ \\end{matrix} \\right.\\ \$]{.math.inline} Find *f(x)* at x = 0, 1, 2, 3, and 4. Solution: Note that each piece of the function in *f(x)* is constant. Then we evaluate: when x = 0, f(0) = 1 when x = 1, f(1) = 2 when x = 2, f(2) = 3 when x = 3, f(3) = 3 when x = 4, f(4) = 3 The domain of the function is the set of real number (x [ ∈  R]{.math.inline}) and the range is y = 1, 2, 3. Example 2: Given the function [\$f\\left( x \\right) = \\left\\{ \\begin{matrix} 1\\ if\\ x \\leq 1 \\\\ 2\\ if\\ 1 \\leq x \\leq 3 \\\\ 3\\ if\\ x \\geq 3 \\\\ \\end{matrix} \\right.\\ \$]{.math.inline} Find *f(x)* at x = 0, 1, 2, 3, 4, 5, and 6. Solution: Note that each piece of the function in *f(x)* is constant. Then we evaluate: when x = 0, f(0) = 1 when x = 1, f(1) = 2 when x = 2, f(2) = 2 when x = 3, f(3) = 2 when x = 4, f(4) = 3 when x = 5, f(5) = 3 when x = 6, f(6) = 3 The domain of the function is the set of real number (x [ ∈  R]{.math.inline}) and the range is y = 1, 2, 3. **E. EVALUATING FUNCTIONS** Example 1: Evaluate the function *f(x)* = 2x + 2 for x = 4 - just replace the variable "x" with "4" Example 2: Evaluate the function *f(x)* = 2x^2^ + 3 for x = 2 - just replace the variable "x" with "2" **F. ALGEBRA OF FUNCTIONS** The operations of addition, subtraction, multiplication, and division of the functions are defined as follows: If *f* and *g* are functions of x is an element of the domain of each function, then (f + g)(x) = f(x) + g(x) (f -- g)(x) = f(x) -- g(x) (f \* g)(x) = f(x) \* g(x) [\$\\left( \\frac{f}{g} \\right)\\left( x \\right) = \\ \\frac{f(x)}{g(x)},\\ g\\left( x \\right) \\neq 0\$]{.math.inline} Example 1: Function f and g given by *f(x)* = 3x-1 and *g(x)* = 2x.Find the following: a. (f + g)(x) b. (f \* g)(x) c. (f -- g)(x) d. [\$\\left( \\frac{f}{g} \\right)\\left( x \\right)\$]{.math.inline} Solution: a. (f + g)(x) = f(x) + g(x) b. (f \* g)(x) = f(x) \* g(x) c. (f -- g)(x) = f(x) -- g(x) d. [\$\\left( \\frac{f}{g} \\right)\\left( x \\right) = \\ \\frac{f(x)}{g(x)}\$]{.math.inline} Example 2: Let *f* and *g* be two functions defined as *f(x)* = 2x+2 and *g(x)* = x + 1. Find the following: a. (f + g) (2) b. (f -- g) (1) c. (f \* g) (1) d. [\$\\left( \\frac{f}{g} \\right)\\left( 2 \\right)\$]{.math.inline} Solution: a. (f + g)(2) = f(2) + g(2) b. (f \* g)(x) = f(x) \* g(x) c. (f -- g)(x) = f(x) -- g(x) d. [\$\\left( \\frac{f}{g} \\right)\\left( x \\right) = \\ \\frac{f(x)}{g(x)}\$]{.math.inline} **G. Composite Functions** *Composition of Function* is another way of combining functions. This method of combining functions uses the output of one function as the input for a second function. In general, given functions *f* and *g*, the composite functions, denoted by *f ○ g* (read as **"f composed of g"**), is defined by (*f ○ g)(x) = f\[g(x)\].* Example: Suppose that *f(x)* = 3x + 4 and *g(x)* = 2x. Find: a. *(f ○ g)(x)* b. *(g ○ f)(x)* c. *(f ○ f)(x)* d. *(g ○ g)(x)* e. *(f ○ g)(2)* f. *(g ○ f)(2)* Solution: a. (f *○ g)(x) = f\[g(x)\]* b. (g *○* f)(x) = g\[f(x)\] c. (f ***○*** f)(x) = f\[f(x)\] d. (g ***○*** g)(x) = g\[g(x)\] e. (f *○ g)(x) = f\[g(x)\]* f. (g *○* f)(x) = g\[f(x)\]