The function $f(x)=x^3+3$ is one-to-one.
a. Find an equation for $f^{-1}$, the inverse function.
b. Verify that your equation is correct by showing that $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$.
a. Select the correct choice below and fill in the answer box(es) to complete your choice.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
A. $f^{-1}(x)=\sqrt[3]{x-3}$, for all $x$
B. $f^{-1}(x)=$, for $x \geq$ $\square$
C. $f^{-1}(x)=$, for $x \leq$ $\square$
D. $f^{-1}(x)=$, for $x \neq$\square
b. Verify that the equation is correct.
$\begin{aligned}
f(f^{-1}(x)) & =f(\square) & \text { and } & f^{-1}(f(x))
\end{aligned}=f^{-1}(\square)=\square$\n$\square$ Substitute.
Simplify.
Answer (2)
Find the inverse function by swapping x and y in the equation y = f ( x ) and solving for y .
The inverse function is f − 1 ( x ) = 3 x − 3 .
Verify the inverse function by showing that f ( f − 1 ( x )) = x and f − 1 ( f ( x )) = x .
The inverse function is verified, and the final answer is f − 1 ( x ) = 3 x − 3 .
Explanation
Understanding the Problem The problem asks us to find the inverse of the function f ( x ) = x 3 + 3 and then verify that the inverse function is correct.
Finding the Inverse Function To find the inverse function, we first replace f ( x ) with y , so we have y = x 3 + 3 . Then we swap x and y to get x = y 3 + 3 . Now we solve for y .
Solving for y Subtracting 3 from both sides gives y 3 = x − 3 . Taking the cube root of both sides gives y = 3 x − 3 . Therefore, the inverse function is f − 1 ( x ) = 3 x − 3 . Since we can take the cube root of any real number, the domain of f − 1 ( x ) is all real numbers.
Verifying the Inverse Function Now we verify that the inverse function is correct by showing that f ( f − 1 ( x )) = x and f − 1 ( f ( x )) = x .
Computing f(f^{-1}(x)) First, we compute f ( f − 1 ( x )) . We have f ( f − 1 ( x )) = f ( 3 x − 3 ) = ( 3 x − 3 ) 3 + 3 = ( x − 3 ) + 3 = x .
Computing f^{-1}(f(x)) Next, we compute f − 1 ( f ( x )) . We have f − 1 ( f ( x )) = f − 1 ( x 3 + 3 ) = 3 ( x 3 + 3 ) − 3 = 3 x 3 = x . Since both compositions give us x , the inverse function is correct.
Final Answer Therefore, the inverse function is f − 1 ( x ) = 3 x − 3 for all x .
Examples
Imagine you have a machine that converts a number into another number by cubing it and adding 3. The inverse function is like having a machine that reverses this process. If you input the result of the first machine into the inverse machine, you get back the original number. This concept is used in cryptography, where encoding and decoding messages rely on inverse functions to ensure secure communication.
The inverse function of f ( x ) = x 3 + 3 is f − 1 ( x ) = 3 x − 3 for all real numbers. Verification shows that both f ( f − 1 ( x )) = x and f − 1 ( f ( x )) = x . Therefore, the chosen option is A.
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