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Chapter 1: Problem 5

Check your answer by evaluating the appropriate function at your answer. Suppose \(f(x)=3 x+2\). Find a formula for \(f^{-1}\).

Short Answer

Expert verified
The inverse function for \(f(x) = 3x + 2\) is \(f^{-1}(x) = \frac{x - 2}{3}\).

Step by step solution

01

Replace \(f(x)\) with \(y\)

Rewrite the equation \(f(x) = 3x + 2\) by replacing \(f(x)\) with \(y\): \[ y = 3x + 2 \]
02

Swap the roles of x and y

Swap x and y in the equation, so \(y\) becomes \(x\), and \(x\) becomes \(y\): \[ x = 3y + 2 \]
03

Solve the equation for \(y\)

Now we want to isolate \(y\) on one side of the equation. Start by subtracting 2 from both sides: \[ x - 2 = 3y \] Now divide both sides by 3 to solve for \(y\): \[ \frac{x - 2}{3} = y \]
04

Replace \(y\) with \(f^{-1}(x)\)

Replace \(y\) with the inverse function notation \(f^{-1}(x)\): \[ f^{-1}(x) = \frac{x - 2}{3} \] So the inverse function for \(f(x) = 3x + 2\) is \(f^{-1}(x) = \frac{x - 2}{3}\). Now, as the exercise asks, you should check your answer by evaluating the original function at the inverse function we found, and vice versa, to see if they return the input value.

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Most popular questions from this chapter

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ Give the table of values for \(f^{-1}\).

Show that the composition of two increasing functions is increasing.

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \(g(x)=f(5 x)\)

Check your answer by evaluating the appropriate function at your answer. Suppose \(f(x)=7 x-5\). Evaluate \(f^{-1}(-3)\).

Suppose you are exchanging cur. rency in the London airport. The currency exchange service there only makes transactions in which one of the two currencies is British pounds, but you want to exchange dollars for Euros. Thus you first need to exchange dollars for British pounds, then exchange British pounds for Euros. At the time you want to make the exchange, the function \(f\) for exchanging dollars for British pounds is given by the formula \(f(d)=0.66 d-1\) and the function \(g\) for exchanging British pounds for Euros is given by the formula \(g(p)=1.23 p-2\) The subtraction of 1 or 2 in the number of British pounds or Euros that you receive is the fee charged by the currency exchange service for each transaction. Is the function describing the exchange of dollars for Euros \(f \circ g\) or \(g \circ f ?\) Explain your answer in terms of which function is evaluated first when computing a value for a composition (the function on the left or the function on the right?).
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