Chapter 8, Problem 14CR

(a)

To determine

The solution of the equation $f(x)=0$, where $f(x)=4x+5$

Answer to Problem 14CR

Solution:

The solution for the equation $f(x)=0$ is $x=-\frac{5}{4}$

Explanation of Solution

Given information:

The function $f(x)=4x+5$

Consider the equation $f(x)=0$

Substitute $f(x)=4x+5$ in the above equation.

  $\Rightarrow f(x)=4x+5=0$

  $\Rightarrow 4x+5=0$ .

Subtract $5$ from both sides.

  $\Rightarrow 4x+5-5=0-5$

  $\Rightarrow 4x=-5$ .

Divide both sides by $4$

  $\Rightarrow \frac{4x}{4}=-\frac{5}{4}$

  $\Rightarrow x=-\frac{5}{4}$ .

Therefore, solution of $f\left(x\right)=0$ is $x=-\frac{5}{4}$

(b)

To determine

The solution of the equation $f(x)=13$, where $f(x)=4x+5$

Answer to Problem 14CR

Solution:

The solution for the equation $f(x)=13$, where $f(x)=4x+5$ is $x=2$

Explanation of Solution

Given information:

The function $f(x)=4x+5$

Consider the equation $f(x)=13$

Substitute $f(x)=4x+5$ in the above equation.

  $\Rightarrow 4x+5=13$

Subtract $5$ from both sides.

  $\Rightarrow 4x+5-5=13-5$

  $\Rightarrow 4x=8$

Divide both sides by $4$

  $\Rightarrow \frac{4x}{4}=\frac{8}{4}$

  $\Rightarrow x=2$ .

Therefore, solution of $f\left(x\right)=13$ is $x=2$

(c)

To determine

To calculate: The solution of the equation $f(x)=g(x)$ where $f(x)=4x+5$ and $g(x)=x^2+5x-24$ .

Answer to Problem 14CR

Solution:

The solution set for the equation $f(x)=g(x)$ is $\left\{ \frac{-1-3\sqrt{13}}{2},\frac{-1+3\sqrt{13}}{2}\right\}$

Explanation of Solution

Given information:

The functions $f(x)=4x+5$ and $g(x)=x^2+5x-24$

Formula used:

Quadratic formula for the zeros of the quadratic equation $a{x}^2+bx+c$

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Calculation:

Consider the equation $f(x)=g(x)$

Solving $f(x)=g(x)$ is equivalent to find the zeros of the function $f(x)-g(x)$ using quadratic formula.

Now, $f(x)-g(x)$

  $f(x)-g(x)= 4x+5-\left(x^2+5x-24\right)$

  $\Rightarrow f(x)-g(x)=4x+5-x^2-5x+24$

  $\Rightarrow f(x)-g(x)=-x^2-x+29$ .

By the quadratic formula, the zeros of the quadratic equation $-x^2-x+29=0$

  $x=\frac{1\pm \sqrt{{\left(-1\right)}^2-4\cdot \left(-1\right)\cdot \left(29\right)}}{2\cdot \left(-1\right)}=\frac{1\pm \sqrt{117}}{-2}=\frac{1\pm 3\sqrt{13}}{-2}$ .

  $\Rightarrow x=\frac{-1-3\sqrt{13}}{2}$ or $x=\frac{-1+3\sqrt{13}}{2}$

Thus, the solution set for the equation $f(x)=g(x)$ is $\left\{ \frac{-1-3\sqrt{13}}{2},\frac{-1+3\sqrt{13}}{2}\right\}$

(d)

To determine

The solution of the inequality $f(x)>0$, where $f(x)=4x+5$

Answer to Problem 14CR

Solution:

The inequality $f(x)>0$, where $f(x)=4x+5$ has the solution $\left(-\frac{5}{4},\infty \right)$

Explanation of Solution

Given information:

The function $f(x)=4x+5$

Consider the inequality $f(x)>0$

  $\Rightarrow 4x+5>0$

Subtract $5$ from both sides of the above inequality to get $4x>-5$

Divide the above inequality by $4$ to get $x>-\frac{5}{4}$

Therefore, the inequality $f(x)>0$ has the solution $\left(-\frac{5}{4},\infty \right)$

(e)

To determine

To calculate: The solution of the inequality $g\left(x\right)\le 0$, where $f(x)=4x+5$

Answer to Problem 14CR

Solution:

The inequality $g\left(x\right)\le 0$ has the solution $\left[-8,3\right]$

Explanation of Solution

Given information:

The function $g(x)=x^2+5x-24$

Consider the inequality $g\left(x\right)\le 0$ .

Substitute $g(x)=x^2+5x-24$ in the above inequality

  $\Rightarrow x^2+5x-24\le 0$

Factor out the above term $x^2+5x-24$

  $\Rightarrow x^2+5x-24=x^2-3x+8x-24$

  $\Rightarrow x^2+5x-24=\left(x+8\right)\left(x-3\right)$

  $\Rightarrow \left(x+8\right)\left(x-3\right)\le 0$ .

From the above, the zeroes of $g(x)=x^2+5x-24$ are $x=-8,3$

Form intervals using zeroes $-8,3$ and check for the solution region of inequality.

  $\begin{array}{cccc}\text{Interval}& (-\infty ,-8)& (-8,3)& (3,\infty )\\ \text{Test}\text{point}& -9& 0& 4\\ \text{Value}\text{of}{x}^2+5x-24& \begin{array}{l}{\left(-9\right)}^2+5\left(-9\right)-24\\ =12\end{array}& \begin{array}{l}{0}^2+5\left(0\right)-24\\ =-24\end{array}& \begin{array}{l}{4}^2+5\left(4\right)-24\\ =12\end{array}\\ \text{Conclusion}& \text{Negative}& \text{Positive}& \text{Negative}\end{array}$

The interval $(-8,3)$ satisfies the inequality $x^2+5x-24\le 0$

To check the inequality at end points.

Substitute $x=-8$ in $x^2+5x-24$ .

  $g\left(-8\right)={\left(-8\right)}^2+5\left(-8\right)-24=0$

Inequality holds at $x=-8$

Substitute $x=3$ in $x^2+5x-24$

  $g\left(3\right)={\left(3\right)}^2+5\left(3\right)-24=0$

Inequality holds at $x=3$

Therefore, the solution for $g\left(x\right)\le 0$ is $\left[-8,3\right]$

(f)

To determine

To graph: The function $f(x)=4x+5$ .

Explanation of Solution

Given information:

The function $f(x)=4x+5$

Graph:

To graph the function $y=f(x)=4x+5$, find at least two points satisfying equation of function

  $\begin{array}{cccc}x& -1& 0& 1\\ f(x)& \begin{array}{l}f(-1)=4(-1)+5\\ =1\end{array}& \begin{array}{l}f(0)=4(0)+5\\ =5\end{array}& \begin{array}{l}f(1)=4(1)+5\\ =9\end{array}\\ (x,f(x))& (-1,1)& (0,5)& (1,9)\end{array}$

Draw these points on a graph and pass a line through these points.

The graph of $y=f\left(x\right)$ is

  

Interpretation:

The graph represents function of $y=f\left(x\right)$

(g)

To determine

To graph: The function $g(x)=x^2+5x-24$

Explanation of Solution

Given information:

The function $g(x)=x^2+5x-24$

Graph:

Consider the function $y=g(x)=x^2+5x-24$

The standard equation $f\left(x\right)=a{x}^2+bx+c$ has its vertex at $\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)$

By comparing with standard equation $a=1,b=5$ and $c=-24$

The equation has vertex at $\left(-\frac{5}{2},f\left(-\frac{5}{2}\right)\right)=\left(-2.5,-30.25\right)$

Also, find some points satisfying equation of function.

  $\begin{array}{ccccccc}x& -9& -7& -5& 0& 2& 4\\ g(x)& 12& -10& -24& -24& -10& 12\\ (x,g(x))& \left(-9,12\right)& \left(-7,-10\right)& \left(-5,-24\right)& \left(0,-24\right)& \left(2,-10\right)& \left(4,12\right)\end{array}$

Draw these points on a graph and pass a curve through these points.

The graph of $y=g\left(x\right)$ is

  

Interpretation:

The graph represents the function $y=g\left(x\right)=x^2+5x-24$