Question

Evaluate the piecewise defined function at the indicated values.

$\begin{array}{l}f\left(x\right)=\left\{\begin{array}{cc}3x& \text{if\hspace{0.33em}}x<0\\ x+1& \text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}}0\le x\le 2\\ {\left(x-2\right)}^2& \text{if\hspace{0.33em}}x>2\end{array}\right.\\ f\left(-5\right),f\left(0\right),f\left(1\right),f\left(2\right),f\left(5\right)\end{array}$

Short answer

The required values are $f\left(-5\right)=-15,f\left(0\right)=1,f\left(1\right)=2,f\left(2\right)=3,f\left(5\right)=9$.

Step-by-step solution

Step 1. Given Information.

The given piecewise function is:

$f\left(x\right)=\left\{\begin{array}{cc}3x& \text{if\hspace{0.33em}}x<0\\ x+1& \text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}}0\le x\le 2\\ {\left(x-2\right)}^2& \text{if\hspace{0.33em}}x>2\end{array}\right.$

Here,$f\left(x\right)=3x$ when $x<0$,$f\left(x\right)=x+1$ when$0\le x\le 2$ and$f\left(x\right)={\left(x-2\right)}^2$ when$x>2$

Step 2. Solve for first value.

Since $-5<0$, therefore substitute$x=-5$ in $f\left(x\right)=3x$.

role="math" localid="1644521695234" $\begin{array}{c}f(-5)=3\left(-5\right)\\ =-15\end{array}$

Step 3. Solve for second value.

Since $0\le 0\le 2$, therefore substitute$x=0$ in $f\left(x\right)=x+1$.

$\begin{array}{c}f\left(0\right)=0+1\\ =1\end{array}$

Step 4. Solve for third value.

Since $0\le 1\le 2$, therefore substitute$x=1$ in $f\left(x\right)=x+1$.

$\begin{array}{c}f\left(1\right)=1+1\\ =2\end{array}$

Step 5. Solve for fourth value.

Since $0\le 2\le 2$, therefore substitute$x=2$ in $f\left(x\right)=x+1$.

$\begin{array}{c}f\left(2\right)=2+1\\ =3\end{array}$

Step 6. Solve for fifth value.

Since $5>2$, therefore substitute $x=5$in $f\left(x\right)={\left(x-2\right)}^2$.

$$\begin{gathered} f\left(5\right)={\left(5-2\right)}^2 \\ ={\left(3\right)}^2 \\ =9 \end{gathered}$$

Thus, the required values are $f\left(-5\right)=-15,f\left(0\right)=1,f\left(1\right)=2,f\left(2\right)=3,f\left(5\right)=9$.