## Precalculus (6th Edition) Blitzer

(a) Consider the function: $f\left( x \right)=2x-1$ Follow the steps to determine the inverse ${{f}^{-1}}\left( x \right)$ of the function $f\left( x \right)=2x-1$. Step 1: Replace the function $f\left( x \right)$ with y in the equation of $f\left( x \right)$. $y=2x-1$ Step 2: Interchange the variables x and y. $x=2y-1$ Step 3: Solve the equation for y. The variable y has to be isolated. Add $1$ to both sides of the equation. So, the equation becomes, $x+1=2y-1+1$ Combine like terms. $x+1=2y$ Divide by $2$ to both the sides of the equation. So, \begin{align} & \frac{x+1}{2}=\frac{2y}{2} \\ & \frac{x+1}{2}=y \\ \end{align} Because the obtained equation defines y as a function of x, then the inverse function exists for the function f. Step 4: Replace y by ${{f}^{-1}}\left( x \right)$ in the equation obtained in step 3. Thus, the inverse function${{f}^{-1}}\left( x \right)$ is obtained as, ${{f}^{-1}}\left( x \right)=\frac{x+1}{2}$ Therefore, the inverse function is ${{f}^{-1}}\left( x \right)=\frac{x+1}{2}$. (b) Given above. (c) The domain and range of the functions in interval notation are: \begin{align} & \text{domain of }f=\text{range of }{{f}^{-1}}=\left( -\infty ,\infty \right) \\ & \text{range of }f=\text{domain of }{{f}^{-1}}=\left( -\infty ,\infty \right) \end{align}