Question

For the function $f\left(x\right)=x^2+5x-2$

Find

$$\begin{gathered} a)f(3\left) \\ b\right)f(-x) \\ c)-f(x\left) \\ d\right)f\left(3x\right) \\ e)\frac{f(x+h)-f\left(x\right)}{h} \end{gathered}$$

Short answer

The solution is

$$\begin{gathered} a)f(3)=22 \\ b)f(-x)=x^2-5x-2 \\ c)-f(x)=-x^2-5x+2 \\ d)f\left(3x\right)=9x^2+15x-2 \\ e)\frac{f(x+h)-f\left(x\right)}{h}=2x+h+5 \end{gathered}$$

Step-by-step solution

Given information

We are given a equation$f\left(x\right)=x^2+5x-2$

Part a) Step 1: Substitute x=3 in the given equation

We get,

$$\begin{gathered} f\left(x\right)=x^2+5x-2 \\ f\left(3\right)=9+15-2 \\ f\left(3\right)=22 \end{gathered}$$

part b) Step 1: To find f(-x) we substitute -x in the function

We get

$$\begin{gathered} f\left(x\right)=x^2+5x-2 \\ f(-x)=x^2-5x-2 \end{gathered}$$

Part c) Step 1: Find -f(x)

We get,

$$\begin{gathered} f\left(x\right)=x^2+5x-2 \\ -f\left(x\right)=-x^2-5x+2 \end{gathered}$$

$$\begin{gathered} f\left(x\right)=x^2+5x-2 \\ -f\left(x\right)=-x^2-5x+2 \end{gathered}$$

part d) Step 1: Find f(3x)

We get,

$$\begin{gathered} f\left(x\right)=x^2+5x-2 \\ f\left(3x\right)=9x^2+15x-2 \end{gathered}$$

Part e) Step 1: Find f(x+h)-f(x)h

We get$$\begin{gathered} \frac{f(x+h)-f\left(x\right)}{h} \\ \frac{{(x+h)}^2+5(x+h)-2-x^2-5x+2}{h} \\ \frac{x^2+2xh+h^2+5x+5h-2-x^2-5x+2}{h} \\ \frac{2xh+h^2+5h}{h} \\ 2x+h+5 \end{gathered}$$$$\begin{gathered} \frac{f(x+h)-f\left(x\right)}{h} \\ \frac{{(x+h)}^2+5(x+h)-2-x^2-5x+2}{h} \\ \frac{x^2+2xh+h^2+5x+5h-2-x^2-5x+2}{h} \\ \frac{2xh+h^2+5h}{h} \\ 2x+h+5 \end{gathered}$$