Question

Suppose that$f\left(x\right)=4x+5\text{and}g\left(x\right)=x^2+5x-24$

(a) Solve$f\left(x\right)=0$

(b) Solve$f\left(x\right)=13$

(c) Solve$f\left(x\right)=g\left(x\right)$

(d) Solve$f\left(x\right)>0$

Short answer

The solution is

$$\begin{gathered} \left(a\right)x=-\frac{5}{4} \\ \left(b\right)x=2 \\ \left(c\right)x=\frac{-1\pm 3\sqrt{13}}{2} \\ \left(d\right)x\in \left(-\frac{5}{4},\infty \right) \end{gathered}$$

Step-by-step solution

Part (a) Step 1. Given Information

We are given two functions,

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

We need to find the solution of$f\left(x\right)=0$

Part (a) Step 2, Solution of the  equation

The given function is,

$f\left(x\right)=4x+5$

For $f\left(x\right)=0,$

$$\begin{gathered} f\left(x\right)=4x+5=0 \\ 4x+5=0 \\ 4x=-5 \\ x=-\frac{5}{4} \end{gathered}$$

Part (b) Step 1. Solution of the equation

The given function is,

$$\begin{gathered} f\left(x\right)=4x+5 \\ forf\left(x\right)=13 \\ 4x+5=13 \\ 4x=13-5 \\ 4x=8 \\ x=2 \end{gathered}$$

Part (c) Step 1. Solution of the equation

The given functions are,$f\left(x\right)=4x+5\text{and}g\left(x\right)=x^2+5x-24$

For $f\left(x\right)=g\left(x\right)$

$$\begin{gathered} 4x+5=x^2+5x-24 \\ {x}^2+x-29=0 \\ U\sin gx=\frac{-b\pm \sqrt{b^2-4ac}}{2a}, \\ =\frac{-1\pm \sqrt{1^2-4\left(1\right)(-29)}}{2\left(1\right)} \\ =\frac{-1\pm \sqrt{1}+116}{2} \\ =\frac{-1\pm \sqrt{117}}{2} \\ =\frac{-1\pm 3\sqrt{13}}{2} \end{gathered}$$

Part (d) Step 1. Solution of the inequality

The given function is $f\left(x\right)=4x+5$

For $f\left(x\right)>0$

$$\begin{gathered} f\left(x\right)=4x+5>0 \\ 4x+5>0 \\ 4x>-5 \\ x>-\frac{5}{4} \\ x\in \left(-\frac{5}{4},\infty \right) \end{gathered}$$