Questions: The function f(x)=4x+7 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)= , for x ≥ B. f^(-1)(x)= for x ≤ C. f^(-1)(x)= for all x D. f^(-1)(x)= for x ≠ b. Verify that the equation is correct f(f^(-1)(x))=f( ) = and f^(-1)(f(x)) =f^(-1)( ) = Substitute. Simplify. The equation is

The function f(x)=4x+7 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)= , for x ≥ B. f^(-1)(x)= for x ≤ C. f^(-1)(x)= for all x D. f^(-1)(x)= for x ≠ b. Verify that the equation is correct f(f^(-1)(x))=f( ) = and f^(-1)(f(x)) =f^(-1)( ) = Substitute. Simplify. The equation is
Transcript text: The function $f(x)=4 x+7$ is one-to-one. a. Find an equation for $f^{-1}$, the inverse function b. Verify that your equation is correct by showing that $f\left(f^{-1}(x)\right)=x$ 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. $f^{-1}(x)=$ $\square$ , for $x \geq$ $\square$ B. $f^{-1}(x)=$ $\square$ for $x \leq$ $\square$ C. $f^{-1}(x)=$ $\square$ for all $x$ D. $f^{-1}(x)=$ $\square$ for $x \neq$ $\square$ b. Verify that the equation is correct $f\left(f^{-1}(x)\right)=f($ $\square$ $=\square$ $\square$ and \[ \begin{aligned} f^{-1}(f(x)) & =f^{-1}(\square) \\ & =\square \end{aligned} \] Substitute. Simplify. The equation is $\square$
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Solution

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Solution Steps

Step 1: Finding the Inverse Function

For a linear function $f(x) = 4x + 7$, solving $y = 4x + 7$ for $x$ gives us $x = \frac{y-7}{4}$.

Hence, $f^{-1}(x) = \frac{x-7}{4}$, which simplifies to ^f^{-1}(x) = (x - 7) / 4^.

Step 2: Verification

Substituting $f^{-1}(x)$ into $f(x)$, we get $f(f^{-1}(x)) = 4*((x - 7) / 4) + 7 = x$, which simplifies to $x$.

Similarly, substituting $f(x)$ into $f^{-1}(x)$, we get $f^{-1}(f(x)) = (f(x) - 7) / 4 = x$, which also simplifies to $x$.

Final Answer:

The inverse function is ^f^{-1}(x) = (x - 7) / 4^.

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