Question

In Problems 76, determine where the graph of $f$is below the graph of $g$by solving the inequality $f\left(x\right)\le g\left(x\right)$. Graph $f$and $g$together.

$$\begin{gathered} f\left(x\right)=x^4 \\ g\left(x\right)=2-x^2 \end{gathered}$$

Short answer

The graph is shown below:

Step-by-step solution

Given information

The given functions are shown below:

$$\begin{gathered} f\left(x\right)=x^4 \\ g\left(x\right)=2-x^2 \end{gathered}$$

Determine the intersection point.

For intersection point,$f\left(x\right)=g\left(x\right)$

role="math" localid="1649315638482" $\begin{array}{rcl}{x}^4& =& 2-x^2\\ x^4+x^2-2& =& 0\\ x^4+2x^2-x^2-2& =& 0\\ x^2\left(x^2+2\right)-1\left(x^2+2\right)& =& 0\\ \left(x^2-1\right)\left(x^2+2\right)& =& 0\\ \left(x-1\right)\left(x+1\right)\left(x^2+2\right)& =& 0\\ x& =& 1,-1\end{array}$

$x^2+2$ gives the imaginary solution.

Graph the function fx

Sketch the graph of $y=f\left(x\right)$having vertex at $(0,0)$and range $[0,\infty )$.

Graph the function gx

Sketch the graph of $y=g\left(x\right)$having vertex at $(0,2)$and range $(-\infty ,2]$.

Determine the shaded portion which satisfy the given inequality.

The portion where $f\left(x\right)=x^4$less than or equal to$g\left(x\right)=2-x^2$is in the interval of $[-1,1]$as shaded in the following figure.

Write the conclusion

The final graph is shown below: