Solutions 5.2: Graphs of Exponential Functions

Solutions to Try Its

1. The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is [latex]y=0[/latex].
Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).

2. The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(3,\infty \right)[/latex]; the horizontal asymptote is y = 3.
Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).

3. [latex]x\approx -1.608[/latex]

4. The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is [latex]y=0[/latex]. 
Graph of the function, f(x) = (1/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).

5. The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is [latex]y=0[/latex].
Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).

6. [latex]f\left(x\right)=-\frac{1}{3}{e}^{x}-2[/latex]; the domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(-\infty ,2\right)[/latex]; the horizontal asymptote is [latex]y=2[/latex].

Solutions to Odd-Numbered Exercises

1. An asymptote is a line that the graph of a function approaches, as x either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

3. [latex]g\left(x\right)=4{\left(3\right)}^{-x}[/latex]; y-intercept: [latex]\left(0,4\right)[/latex]; Domain: all real numbers; Range: all real numbers greater than 0.

5. [latex]g\left(x\right)=-{10}^{x}+7[/latex]; y-intercept: [latex]\left(0,6\right)[/latex]; Domain: all real numbers; Range: all real numbers less than 7.

7. [latex]g\left(x\right)=2{\left(\frac{1}{4}\right)}^{x}[/latex]; y-intercept: [latex]\left(0,\text{ 2}\right)[/latex]; Domain: all real numbers; Range: all real numbers greater than 0.

9. y-intercept: [latex]\left(0,-2\right)[/latex]
Graph of two functions, g(-x)=-2(0.25)^(-x) in blue and g(x)=-2(0.25)^x in orange.

11.
Graph of three functions, g(x)=3(2)^(x) in blue, h(x)=3(4)^(x) in green, and f(x)=3(1/4)^(x) in orange.

13. B

15. A

17. E

19. D

21. C

23.
Graph of two functions, f(x)=(1/2)(4)^(x) in blue and -f(x)=(-1/2)(4)^x in orange.

25.
Graph of two functions, -f(x)=(4)(2)^(x)-2 in blue and f(x)=(-4)(2)^x+1 in orange.

27. Horizontal asymptote: [latex]h\left(x\right)=3[/latex]; Domain: all real numbers; Range: all real numbers strictly greater than 3.
Graph of h(x)=2^(x)+3.

29. As [latex]x\to \infty[/latex] , [latex]f\left(x\right)\to -\infty\\[/latex] ;

As [latex]x\to -\infty[/latex] , [latex]f\left(x\right)\to -1[/latex]

31. As [latex]x\to \infty\\[/latex] , [latex]f\left(x\right)\to 2[/latex] ;

As [latex]x\to -\infty[/latex] , [latex]f\left(x\right)\to \infty[/latex]

33. [latex]f\left(x\right)={4}^{x}-3[/latex]

35. [latex]f\left(x\right)={4}^{x - 5}[/latex]

37. [latex]f\left(x\right)={4}^{-x}[/latex]

39. [latex]y=-{2}^{x}+3[/latex]

41. [latex]y=-2{\left(3\right)}^{x}+7[/latex]

43. [latex]g\left(6\right)=800+\frac{1}{3}\approx 800.3333[/latex]

45. [latex]h\left(-7\right)=-58[/latex]

47. [latex]x\approx -2.953[/latex]

49. [latex]x\approx -0.222[/latex]

51. The graph of [latex]G\left(x\right)={\left(\frac{1}{b}\right)}^{x}[/latex] is the reflection about the y-axis of the graph of [latex]F\left(x\right)={b}^{x}[/latex]; For any real number [latex]b>0[/latex] and function [latex]f\left(x\right)={b}^{x}[/latex], the graph of [latex]{\left(\frac{1}{b}\right)}^{x}[/latex] is the the reflection about the y-axis, [latex]F\left(-x\right)[/latex].

53. The graphs of [latex]g\left(x\right)[/latex] and [latex]h\left(x\right)[/latex] are the same and are a horizontal shift to the right of the graph of [latex]f\left(x\right)[/latex]; For any real number n, real number [latex]b>0[/latex], and function [latex]f\left(x\right)={b}^{x}[/latex], the graph of [latex]\left(\frac{1}{{b}^{n}}\right){b}^{x}[/latex] is the horizontal shift [latex]f\left(x-n\right)[/latex].