Question

In Problems 75 , 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.

$$\begin{gathered} f\left(x\right)=x^4-1 \\ g\left(x\right)=x-1 \end{gathered}$$

Short answer

The graph is shown below:

Step-by-step solution

Given information

The given functions are shown below:

$$\begin{gathered} f\left(x\right)=x^4-1 \\ g\left(x\right)=x-1 \end{gathered}$$

Determine the intersection point. 

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

$$\begin{gathered} x^4-1=x-1 \\ {x}^4-x=0 \\ x\left(x^3-1\right)=0 \\ x\left(x-1\right)\left(x^2+1+x\right)=0 \\ x=0,1 \end{gathered}$$

$x^2+x+1$gives the imaginary solution.

Graph the function fx

Sketch the graph of $y=f\left(x\right)$having vertex at $(0,-1)$and range $[-1,\infty )$.

Graph the function gx

Sketch the graph of $y=g\left(x\right)$having x-intercept and y-intercept at $(1,0)$and $\left(0,-1\right)$and range $(-\infty ,+\infty )$.

Determine the shaded portion which satisfy the given inequality. 

The portion where $f\left(x\right)=x^4-1$less than or equal to $g\left(x\right)=x-1$is in the interval of $[0,0]\mathrm{to}[1,0]$ as shaded in the following figure.

Write the conclusion

The final graph is shown below: