Question

The graph of f and g are given. Find a formula for the function g.

Short answer

$g\left(x\right)=-\sqrt{x+2}$

Step-by-step solution

Step 1. Concept of reflection and Horizontal shifting.

To graph $y=f(x-c)$, shifts the graph$y=f\left(x\right)$ to the right c units

And, to graph$y=f(x+c)$ shifts the graph$y=f\left(x\right)$ to the left c units

And,

To graph $y=-f\left(x\right)$, reflect the graph of $y=f\left(x\right)$ in the x-axis

To graph $y=f(-x)$, reflect the graph of $y=f\left(x\right)$ in the y-axis

Step 2. Given condition.

Given equation is

$f\left(x\right)=\sqrt{x}$; And the graph of$g\left(x\right)$ is

Step 3. Determining the formula for the function g.

The graph first reflect in the$x$-axisthen, shift 2 unit left

So,$$\begin{gathered} c=2 \\ g\left(x\right)=-\sqrt{x+2} \end{gathered}$$