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Chapter 2 Nonlinear Functions
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Properties of Functions
Section 2.1 Properties of Functions
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Figure 5a - 5b
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Figure 5c - 5d
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Figure 6
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Your Turn 1 Find the domain and range for the function Solution: The domain includes only those values of x satisfying since the denominator cannot be zero. Using the methods for solving a quadratic inequality produces the domain Because the numerator can never be zero, the denominator can take on any positive real number except for 0, allowing y to take on any positive value except for 0, so the range is
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Figure 7
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Your Turn 2 Given the function find each of the following. (a) (b) All values of x such that (a) Solution: Replace x with the expression x + h and simplify. (b) Solution: Set f (x) equal to − 5 and then add 5 to both sides to make one side equal to 0. Continued
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Your Turn 2 continued This equation does factor as Set each factor equal to 0 and solve for x.
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Quadratic Functions; Translation and Reflection
Section 2.2 Quadratic Functions; Translation and Reflection
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Figure 18
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Your Turn 1 For the function (a) complete the square, (b) find the y-intercept, (c) find the x intercepts, (d) find the vertex, and (e) sketch the graph. Solution (a): To begin, factor 2 from the x-terms so the coefficient of x2 is 1: Next, we make the expression inside the parentheses a perfect square by adding the square of one-half of the coefficient of x, Continued
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Your Turn 1 continued Solution (b):The y-intercept (where x = 0) is − 1. Solution (c): To find the x-intercepts, solve Use the quadratic formula to verify that the x-intercepts are at Solution (d): The function is now in the form Continued
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Your Turn 1 continued
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Figure 19
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Figure 21-22
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Figure 23-25
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Figure
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Figure
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Figure 32
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Polynomial and Rational Functions
Section 2.3 Polynomial and Rational Functions
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Your Turn 1 Graph Solution: Using the principles
of translation and reflection, we recognize that this is similar to the graph of but reflected vertically (because of the negative in front of x6 ) and 64 units up.
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Figure
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Exponential Functions
Section 2.4 Exponential Functions
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Figure 48
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Your Turn 1 Solve Solution: Since the bases must be the same, write 25 as 52 and 125 as 53.
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Your Turn 2 Find the interest earned on $4400 at 3.25% interest compounded quarterly for 5 years. Solution: Use the formula for compound interest with P = 4400, r = , m = 4, and t = 5. The investment plus the interest is $ The interest amounts to $ − $4400 = $
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Your Turn 3 Find the amount after 4 years if $800 is invested in an account earning 3.15% compounded continuously. Solution: In the formula for continuous compounding, let P = 800, t = 4 and r = to get or $
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Figure
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Figure 53
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Logarithmic Functions
Section 2.5 Logarithmic Functions
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Example 1
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Figure 54
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Example 3 If all the following variable expressions represent positive numbers, then for a > 0, a ≠ 1, the statements in (a)–(c) are true.
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Your Turn 3 Write the expression as a sum, difference, or product of simpler logarithms. Solution: Using the properties of logarithms,
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Figure 55
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Your Turn 4 Evaluate Solution: Using the change-of-base theorem for logarithms with x = 50 and a = 3 gives
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Your Turn 5 Solve for x: Solution: This leads to two solutions: x = − 4 and x = 2. But notice that x = − 4 is not a valid value for x in the original equation, since the logarithm of a negative number is undefined. The only solution is, therefore, x = 2.
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Your Turn 6 Solve for x: Solution: Taking natural logarithms on both sides gives
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Figure 56
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Applications: Growth and Decay; Mathematics of Finance
Section 2.6 Applications: Growth and Decay; Mathematics of Finance
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Your Turn 1 Yeast in a sugar solution is growing at a rate such that 5 g grows exponentially to 18 g after 16 hours. Find the growth function, assuming exponential growth. Solution: The values of y0 and k in the exponential growth function y = y0 e kt must be found. Since y0 is the amount present at time t = 0, y0 = 5. To find k, substitute y = 18, t = 16, and y0 = 5 into the equation y = y0 e kt . Now take natural logarithms on both sides and use the power rule for logarithms and the fact that Continued
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Your Turn 1 continued The exponential growth function is where y is the number of grams of yeast present after t hours.
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Graphs of Basic Functions
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