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Chapter 3: Problem 58

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=3 \cdot 4^{x}-5 $$

Short Answer

Expert verified
The inverse function of \(f(x) = 3 \cdot 4^x - 5\) is \(f^{-1}(x) = \log_4 \left(\frac{x + 5}{3}\right)\).

Step by step solution

01

Rewrite the function with y

Rewrite the function with y instead of \(f(x)\). This will help us understand the relationship between x and y, which we will then use to find the inverse function: $$ y = 3 \cdot 4^x - 5 $$
02

Solve for x

In order to find the inverse function, we now want to switch the roles of x and y to solve for x in terms of y. That means swapping x and y in the equation we obtained in Step 1: $$ x = 3 \cdot 4^y - 5 $$
03

Isolate the exponential term

We need to isolate the exponential term containing y. Begin by adding 5 to both sides of the equation: $$ x + 5 = 3 \cdot 4^y $$
04

Divide both sides by 3

Next, divide both sides of the equation by 3 to further isolate the exponential term: $$ \frac{x + 5}{3} = 4^y $$
05

Apply the logarithm

Now, it's time to eliminate the exponent. To do this, we need to apply the logarithm with base 4 to both sides of the equation: $$ \log_4 \left(\frac{x + 5}{3}\right) = y $$
06

Rewrite the inverse function

Finally, write the inverse function \(f^{-1}(x)\) using the result from Step 5: $$ f^{-1}(x) = \log_4 \left(\frac{x + 5}{3}\right) $$ So, the inverse function of \(f(x) = 3 \cdot 4^x - 5\) is \(f^{-1}(x) = \log_4 \left(\frac{x + 5}{3}\right)\).

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