Precalculus by Richard Wright

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3-01 Exponential Functions

Credit card companies charge interest every day. Banks pay interest every month. How much money would be owed after several years? Or how much money is in a bank account after 10 years? These questions can be answered using exponential functions.

Evaluate Exponential Functions

Repeated addition is multiplication. For example, 4 + 4 + 4 + 4 + 4 = 5(4). Repeated multiplication is exponents. For example, 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 = 4⁵. An exponential function is a function that has the variable in the exponent.

Graph Exponential Functions

The graphs of exponential functions have a characteristic ever-changing curve as in .

Notice the domain is all real numbers because any real number can be an exponent. The range is y > 0 if a > 0; and y < 0 if a < 0. The y-intercept is (0, a), and horizontal asymptote is y = 0.

Graph an Exponential Function

Graph

1. f(x) = 3^x
2. g(x) = 5^x

Are these exponential growth or decay?

Solution

Since b > 0 in both cases, this is exponential growth. Notice the graph increases.

Graph an Exponential Function

Graph

1. $f\left(x\right)={\left(\frac{1}{3}\right)}^x$
2. $g\left(x\right)={\left(\frac{1}{5}\right)}^x$

Are these exponential growth or decay?

Solution

Since b < 0 in both cases, this is exponential decay. Notice the graph decreases.

Graph $h\left(x\right)=\left(\frac{1}{2}\right)3^x$

Answer

Notice the graphs of exponential growth functions like those in Example 2 always increase. The graphs of exponential decay functions like those in Example 3 always decrease. Because the graphs never switch from increasing to decreasing or vise versa, each input corresponds to a unique output. This type of function is called one-to-one. Each x gives a unique y-value that is not paired with another x.

Exponential functions follow the rules of transformations as given in Lesson 1-07.

Describe the transformation and graph the following function: f(x) = −(2)^{x − 2} − 1.

Answer

Reflected over the x-axis, shifted right 2 and down 1.

The Natural Base e

Many uses of exponential functions are simplest using the number e ≈ 2.718281828…. This number is called the natural base and is found by letting n get very large and approach ∞ in the expression ${\left(1+\frac{1}{n}\right)}^n$. The natural base, e, can be useful in calculus because the slope of the function f(x) = e^x is also e^x. For example, the slope of the graph of f(x) = e^x at x = 2 is e².

Graph an Exponential Function with Base e

Graph $f\left(x\right)=\frac{1}{2}{e}^x-1$.

Solution

Applications

When a credit card company or bank calculates, or compounds, interest, they quote a yearly interest rate. But the banks then calculate interest monthly or daily. To derive a formula for calculating compound interest, look at the pattern. In the following table P is the principle and r is the yearly interest rate (APR). Each year's interest is calculated by Pr. The amount in the account will be P + Pr which is the principle plus the interest. This factors to P(1 + r).

If the interest is compounded n times per year, the formula needs to be modified to $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

Now think about what happens as the number of compoundings per year gets very large. The most often interest can be compounded is continuously where the interest is calculated every instant when n = ∞. To derive the formula for continuously compounded interest, let m = ⁿ⁄_r, so n = mr.

$$\begin{array}{c}A=P{\left(1+\frac{r}{mr}\right)}^{mrt}\\ A=P{\left(1+\frac{1}{m}\right)}^{mrt}\\ A=P{\left({\left(1+\frac{1}{m}\right)}^m\right)}^{rt}\end{array}$$

For continuous compounding, m becomes large and approaches ∞. The expression in parentheses is the definition of e when m approaches ∞.

A = Pe^rt

Practice Exercises

1. Two types of fish have been introduced into a pond. The population of fish A can be modeled by A(t) = 20(1.05)^t and fish B by B(t) = 40(1.025)^t where t is in years. (Round your answers to the nearest whole number.)
2. Which fish population is growing at a faster rate?
3. What type of fish was initially introduced in greater quantity?
4. Assuming the models are still accurate, which type of fish will have the bigger population after 10 years?
5. Evaluate the function for the given values.
6. f(x) = 2⋅3^x
   
   1. f(0)
   2. f(2)
   3. f(−1)
   4. f(1⁄2)
7. f(x) = −e^{x + 1}
   
   1. f(0)
   2. f(2)
   3. f(−1)
   4. f(1⁄2)
8. What is an asymptote?
9. The graph of f(x) = 2^x is reflected over the x-axis and stretched vertically by a factor of 5. (a) Write the new function g(x), and (b) State its y-intercept, domain, and range.
10. Write a function that represents the transformation of f(x) = 3^x.
11. Shift f(x) 3 units right and 2 units down
12. Reflect f(x) over the y-axis and shift 2 units up
13. The graph is a transformation of y = 2^x. Write an equation describing the transformation.

14. 

15. Graph the function. Then state the domain, range, asymptote, and whether it is exponential growth, decay, or neither.
16. f(x) = 3^{x − 2}
17. $g\left(x\right)={\left(\frac{1}{2}\right)}^x-3$
18. h(x) = 2^{1 − x} + 1
19. j(x) = −e^x + 2
20. *k(x) = 2e^x
21. Sally invests $1500 in an account that pays 7.5% interest. How much is the account worth after 20 years if the interest is compounded
    
    1. annually?
    2. semiannually?
    3. monthly?
    4. daily?
    5. continuously?
22. Mixed Review
23. (2-09) Solve $\frac{x+1}{x}\le 0$.
24. (2-08) Find the asymptotes and graph $f\left(x\right)=\frac{x+1}{x}$.
25. (2-05) Find the rational zeros of x⁴ − 2x³ − 11x² + 6x + 24.
26. (2-06) Find the irrational zeros of x⁴ − 2x³ − 11x² + 6x + 24.
27. (1-09) Verify that f(x) = 2x + 3 and $g\left(x\right)=\frac{x-3}{2}$ are inverses by finding their composition.

Answers

1. Fish A has a larger base (1.05).
2. Fish B had 20 more than fish A.
3. Fish A will have 33 fish and B will have 51 fish.
4. 2; 18; 2⁄3; $2\sqrt{3}$
5. −2.718; −20.086; −1; −4.482
6. An asymptote is a line that the graph of a function approaches, as x either increases or decreases without bound or approaches from the left or right. 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.
7. g(x) = −5(2)^x; y-int: (0, −5); domain: all real numbers; range: (−∞, 0)
8. g(x) = 3^{x − 3} − 2
9. g(x) = 3^−x + 2
10. y = −2^x + 2

11. 

    ; domain: all real numbers; range: (0, ∞); y = 0; growth

12. 

    ; domain: all real numbers; range: (−3, ∞); y = −3; decay

13. 

    ; domain: all real numbers; range: (1, ∞); y = 1; decay

14. 

    ; domain: all real numbers; range: (−∞, 2); y = 2; neither

15. 

    ; domain: all real numbers; range: (0, ∞); y = 0; growth
16. $6371.78; $6540.57; $6691.23; $6721.50; $6722.53
17. [−1, 0)
18. VA: x = 0; HA: y = 1;

    

19. −2, 4
20. $\pm \sqrt{3}$
21. f(g(x)) = x