Chapter 4.5, Problem 59E

To determine

To solve: the equation $f\left(x\right)=g\left(x\right)$ for the given function graphically and algebraically.

Answer to Problem 59E

  $x=1$

Explanation of Solution

Given information:

Given functions are,

  $f\left(x\right)=x^3+x^2-x-1$

And

  $g\left(x\right)=-x+1$

Calculation:

Consider the given functions.

  $f\left(x\right)=x^3+x^2-x-1$   ...... (1)

  $g\left(x\right)=-x+1$   ...... (2)

Now, the points of intersection of the graph of $f\left(x\right)$ and $g\left(x\right)$ will be the solution of $f\left(x\right)=g\left(x\right)$ .

Construct the graphs of $f\left(x\right)$ and $g\left(x\right)$ in same set of axes.

  

Here, the point of intersection is $\left(1,0\right)$ .

Thus, the solution is $x=1$ .

Now, solve $f\left(x\right)=g\left(x\right)$ algebraically.

  $\begin{array}{c}{x}^3+x^2-x-1=-x+1\\ x^3+x^2-x-1+x-1=0\\ x^3+x^2-2=0\\ x^3-x^2+2x^2-2=0\\ x^2\left(x-1\right)+2\left(x-1\right)=0\\ \left(x-1\right)\left(x^2+2\right)=0\end{array}$

Now, $x^2+2\ne 0$ for any real x .

So, $x-1=0$

Hence, $x=1$ is the only real solution of $f\left(x\right)=g\left(x\right)$ .