Lesson 20

Whenever the input of a function \(f\) increases by 1, the output increases by 5. Which of these equations could define \(f\)?

\(f(x) = 3x + 5\)

\(f(x) = 5x + 3\)

\(f(x) = 5^x\)

\(f(x) = x^5\)

The function \(f\) is defined by \(f(x) = 2^x\). Which of the following statements is true about the values of \(f\)? Select all that apply. 

When the input \(x\) increases by 1, the value of \(f\) increases by 2.

When the input \(x\) increases by 1, the value of \(f\) increases by a factor of 2.

When the input \(x\) increases by 3, the value of \(f\) increases by 8.

When the input \(x\) increases by 3, the value of \(f\) increases by a factor of 8.

When the input \(x\) increases by 4, the value of \(f\) increases by a factor of 4.

The two lines on the coordinate plane are graphs of functions \(f\) and \(g\).

1. Use the graph to explain why the value of \(f\) increases by 2 each time the input \(x\) increases by 1. 
2. Use the graph to explain why the value of \(g\) increases by 2 each time the input \(x\) increases by 1.

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The function \(h\) is given by \(h(x) = 5^x\).

1. Find the quotient \(\frac{h(x+2)}{h(x)}\).
2. What does this tell you about how the value of \(h\) changes when the input is increased by 2?
3. Find the quotient \(\frac{h(x+3)}{h(x)}\).
4. What does this tell you about how the value of \(h\) changes when the input is increased by 3?

For each of the functions \(f, g, h, p,\) and \(q\), the domain is \(0 \leq x \leq 100\). For which functions is the average rate of change a good measure of how the function changes for this domain? Select all that apply. 

\(f(x) = x + 2\)

\(g(x) = 2^x\)

\(h(x) = 111x - 23\)

\(p(x) = 50,\!000 \boldcdot 3^{x}\)

\(q(x) = 87.5\)

The average price of a gallon of regular gasoline in 2016 was $2.14. In 2017, the average price was $2.42 a gallon—an increase of 13%.

At that rate, what will the average price of gasoline be in 2020?

A credit card charges a 14% annual nominal interest rate and has a balance of $500.

If no payments are made and interest is compounded quarterly, which expression could be used to calculate the account balance, in dollars, in 3 years?

\(500\boldcdot\left(1 + 0.14\right)^3\)

\(500\boldcdot\left(1 + \frac{0.14}{4}\right)^3\)

\(500\boldcdot\left(1 + \frac{0.14}{4}\right)^{12}\)

\(500\boldcdot\left(1+ \frac{0.14}{4}\right)^{48}\)

Here are equations that define four linear functions. For each function, write a verbal description of what is done to the input to get the output, and then write the inverse function.

1. \(a(x)=x-4\)
2. \(b(x)=2x-4\)
3. \(c(x)=2(x-4)\)
4. \(d(x)= \frac{x}{4}\)