Nile University MATH100 Precalculus PDF

MATH100 Precalculus Lecture #1 Introduction to functions Solving Inequalities Domain and Range of a Function 1  Real Numbers The set of real numbers is the set of a...

MATH100 Precalculus Lecture #1 Introduction to functions Solving Inequalities Domain and Range of a Function 1  Real Numbers The set of real numbers is the set of all rational and irrational numbers 𝒑 𝟏 Rational Numbers 𝑸 = 𝒒 ฬ 𝒑, 𝒒 ∈ 𝒁, 𝒒 ≠ 𝟎 , 𝑸 = { … , − , 𝟎 … } 𝟐 Integers Irrational Numbers 𝟑 𝒁 = {.. , −𝟐, −𝟏, 𝟎, 𝟏, 𝟐, … } 𝑸′ = { … , 𝝅, 𝒆, 𝟑, −𝟓, … } Natural Numbers 𝑵 = {𝟏, 𝟐, 𝟑, … } 𝑹: Real Numbers Venn Diagram 2  Solving Inequalities: Example 1 Example Solve the following inequality: 2𝑥 − 4 > 8 Solution 2𝑥 − 4 > 8 2𝑥 > 8 + 4 2𝑥 > 12 𝑥>6 The solution set is 𝒙 ∈ (𝟔, ∞) 3  Solving Inequalities: Example 2 Example Solve the following inequality: 2𝑥 + 1 < −9 Solution 2x + 1  − 9 2x  − 10 x  −5 The solution set is 𝒙 ∈ (−∞, −𝟓) 4  Solving Inequalities: Example 3 Example Solve the following inequality: 1 − 2𝑥 < 9 Solution 5  Solving Inequalities: Example 4 Example Solve the following inequality 2𝑥 2 − 3𝑥 + 1 ≤ 0 Solution First, find the zeros by solving the equation 2 x 2 − 3x + 1 = 0 2 x 2 − 3x + 1 = 0 (2 x − 1)(x − 1) = 0 2 x − 1 = 0 or x − 1 = 0 1 x = or x = 1 2 6  Solving Inequalities: Example 4 Continued Now consider the intervals around the zeros and test a value from each interval in the inequality. The intervals can be seen by putting the zeros on a number line. 1/2 1 7  Solving Inequalities: Example 4 Continued Interval Test Point Substitute in Inequality 2𝑥 2 − 3𝑥 + 1 ≤ 0 True/False 2𝑥 2 − 3𝑥 + 1 ≤ 0  1  − ,  x=0 2(0) − 3(0) + 1 = 0 − 0 + 1 = 1  0 2 False  2 2𝑥 2 − 3𝑥 + 1 ≤ 0 −1 2 1  3 3 3 9 9  ,1 x= 2  − 3  + 1 = − + 1 = 0 True 2  4   4   4 8 4 8 2𝑥 2 − 3𝑥 + 1 ≤ 0 (1,  ) x=2 2(2) − 3(2) + 1 = 8 − 6 + 1 = 3  0 2 False 8  Solving Inequalities: Example 4 Continued 1 Thus, the interval [ , 1] makes up the solution set for the inequality. 2 2 x 2 − 3x + 1  0 9  Absolute Value Inequalities Case 1 𝑥 < 𝑎 , 𝑎 > 0 is equivalent to −𝑎 < 𝑥 < 𝑎 10  Absolute Value Inequalities Case 2 𝑥 > 𝑎 , 𝑎 > 0 is equivalent to 𝑥 < −𝑎 OR 𝑥>𝑎 11  Absolute Value Inequalities – Example 1 Example Solve the following inequality x−3  5 Solution −5  x − 3  5 −2  x  8 x  (−2,8) 12  Absolute Value Inequalities – Example 2 Example Solve the following inequality 2 x + 1  9 Solution 2x + 1  − 9 OR 2x + 1  9 2x  − 10 2x  8 x  −5 x4 𝒙 ∈ (−∞, −𝟓) ∪ (𝟒, ∞) 13  Absolute Value Inequalities – Example 3 Example Solve the following inequality 3𝑥 − 2 + 3 ≥ 11 Solution 3x − 2  8 3𝑥 − 2 ≥ 8 OR 3𝑥 − 2 ≤ −8 3𝑥 ≥ 10 OR 3𝑥 ≤ −6 10 𝑥≥ OR 𝑥 ≤ −2 3 10 𝑥 ∈ (−∞, −2ሿ ∪ ቈ , ∞) 3 -2 3 4 14  Function Definition A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. 𝒚 = 𝒇(𝒙) Examples 𝒚 = 𝒙𝟐 𝒚= 𝒙 15  Function Definition Examples Not a Function 𝑥 2+ 𝑦 2= 𝑎2 16  The Vertical Line Test of a Function No vertical line can intersect the graph of a function more than once 𝑥 2+ 𝑦 2= 𝑎2 17 The Vertical Line Test of a Function Hint: Pass a pencil across the graph held vertically to represent a vertical line. The pencil crosses the graph more than once. This is not a function because there are two 𝑦-values for the same 𝑥-value. 18  Domain and Range of a Function Domain of the function 𝒚 = 𝒇(𝒙) The set of all possible values of 𝒙 Range of the function 𝒚 = 𝒇(𝒙) The set of all values of 𝒚 when 𝒙 is in the domain Symbols of Domain and Range Domain of 𝒇 𝒙 𝐢𝐬 𝑫(𝒇) Range of 𝒇 𝒙 is 𝑹(𝒇) A diagram showing a function as a machine 19 Why aren't we allowed to divide by Zero? A US Navy warship Yorktown 20  Why aren't we allowed to divide by Zero? In September of 1997, the USS Yorktown, a US Navy warship, experienced "an engineering local area network casualty" during maneuvers off the coast of Virginia. The missile cruiser, a prototype for the Navy's PC-based Smart Ship program, had quite suddenly become dead in the water, with its propulsion systems rendered useless. The failure was due to a human crew member entering the number 0 into a database entry field. This led to the computer attempting to divide by 0 and crashing as the result of a subsequent buffer overrun, a computing error in which a system begins overwriting already-allocated memory slots, with all sorts of possibly undesirable outcomes, such as rendering a warship useless. The error spread throughout the network. https://motherboard.vice.com/en_us/article/3dk58b/why-were-not- allowed-to-divide-by-zero 21  Domain of a function: Examples Example 1 Find the domain of the function 𝑔 𝑥 = −3𝑥 2 + 4𝑥 + 5 Solution Domain is 𝑅 (all real numbers) 22  Domain of a function: Examples 𝑥−4 Example 2 Find the domain of the function 𝑓 𝑥 = 𝑥+3 Solution 𝑥+3≠0 𝑥 ≠ −3 Domain is 𝑅 − {− 3} (All real numbers except − 3) 23  Domain of a function: Examples 𝑥+1 Example 3 Find the domain of the function 𝑓 𝑥 = 𝑥−3 (𝑥+5) Solution Domain of 𝑓 𝑥 = 𝑅 − {3, −5} (All real numbers except 3, − 5) 24  Domain of a function: Examples 4 Example 4 Find the domain of the function 𝑓 𝑥 = 𝑥 2 −9 Solution 4 4 𝑓 𝑥 = = 𝑥 2 −9 𝑥−3 𝑥+3 Domain of 𝑓 𝑥 = 𝑅 − {3, −3} 25  Domain of a function: Examples 𝑥−1 Example 5 Find the domain of the function 𝑓 𝑥 = 𝑥 2 +4𝑥+3 Solution 𝑥−1 𝑥−1 𝑓(𝑥) = 2 = 𝑥 + 4𝑥 + 3 (𝑥 + 1)(𝑥 + 3) 𝐷 = 𝑅 − {−1, −3} 26  Domain of a function: Examples 4 Example 6 Find the domain of the function 𝑓 𝑥 = 𝑥 2 +9 Solution Domain of 𝑓 𝑥 = 𝑅 because 𝑥 2 + 9 ≠ 0 27  Domain of a function: Examples 𝑥 Example 7 Find the domain of the function 𝑓 𝑥 = 𝑥 +1 Solution Domain of 𝑓 𝑥 = 𝑅 because 𝑥 + 1 ≠ 0 28  Domain of a function: Examples 𝑥 Example 8 Find the domain of the function 𝑓 𝑥 = 𝑥 −1 Solution Domain of 𝑓 𝑥 = 𝑅 − {1, −1} 29 Finding the domain and range of a function from its graph 30 Finding the domain and range of a function from its graph 31 Finding the domain and range of a function from its graph - Examples Example Find the domain and range f the function 𝑓 𝑥 = 𝑥 + 3 from its graph y Range (–3, 0) 1 x –1 Domain Solution The domain is [–3,∞). The range is [0,∞). 32  Finding the domain and range of a function - Examples Example 1 Find the domain and range of the function 𝑓 𝑥 = 4 − 3𝑥 Solution 4 − 3x  0 −3x  −4 x  4 3 The domain is (–∞, 4/3]. The range is [0,∞). 33  Finding the domain and range of a function - Examples 1 Example 2 Find the domain and range of the function 𝑓 𝑥 = 4−𝑥 2 Solution We have to set 4 − 𝑥 2 > 0 𝑥