Question

the function f(x)=2x + 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)=, for all x
○b. f^(-1)(x)=, for x≥
○c. f^(-1)(x)=, for x≠
○d. f^(-1)(x)=, for x≤

Explanation:

Step1: Set \(y = f(x)\)

Let \(y = 2x+3\).

Step2: Solve for \(x\) in terms of \(y\)

Subtract 3 from both sides: \(y - 3=2x\). Then divide by 2: \(x=\frac{y - 3}{2}\).

Step3: Replace \(x\) with \(f^{-1}(x)\) and \(y\) with \(x\)

\(f^{-1}(x)=\frac{x - 3}{2}\), and the domain of \(f^{-1}(x)\) is all real - numbers. So \(f^{-1}(x)=\frac{x - 3}{2}\) for all \(x\).

Step4: Verify \(f(f^{-1}(x))=x\)

Substitute \(f^{-1}(x)\) into \(f(x)\): \(f(f^{-1}(x))=2(\frac{x - 3}{2})+3=x - 3+3=x\).

Step5: Verify \(f^{-1}(f(x))=x\)

Substitute \(f(x)\) into \(f^{-1}(x)\): \(f^{-1}(f(x))=\frac{(2x + 3)-3}{2}=\frac{2x}{2}=x\).

Answer:

A. \(f^{-1}(x)=\frac{x - 3}{2}\), for all \(x\)