Question

the following function is one - to - one. find the inverse of the function and graph the function and its inverse on the same set of axes.\\(f(x)=2x + 3\\)\\(f^{-1}(x)=square\\)\\(\text{(type a simplified fraction.)}\\)\\(\text{use the graphing tool to graph the function and its inverse.}\\)\\(\text{click to enlarge graph}\\)

Explanation:

Step1: Replace f(x) with y

We start with the function \( f(x) = 2x + 3 \). Replace \( f(x) \) with \( y \), so we have \( y = 2x + 3 \).

Step2: Swap x and y

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 2y + 3 \).

Step3: Solve for y

Now we solve the equation \( x = 2y + 3 \) for \( y \). First, subtract 3 from both sides: \( x - 3 = 2y \). Then, divide both sides by 2: \( y=\frac{x - 3}{2}\), which can be rewritten as \( y=\frac{1}{2}x-\frac{3}{2} \).

Step4: Replace y with \( f^{-1}(x) \)

Finally, we replace \( y \) with \( f^{-1}(x) \) to get the inverse function. So \( f^{-1}(x)=\frac{x - 3}{2} \) (or \( \frac{1}{2}x-\frac{3}{2} \)).

Answer:

\( f^{-1}(x)=\frac{x - 3}{2} \) (or \( \frac{1}{2}x-\frac{3}{2} \))