Q. 39 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 39– 46, find the following for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

$f\left(x\right)=3x^2+2x-4$

Step-by-Step Solution

Answer

The value of

$$\begin{gathered} f\left(0\right)=-4 \\ f\left(1\right)=1 \\ f(-1)=-3 \\ f(-x)=3x^2-2x-4 \\ -f\left(x\right)=-3x^2-2x+4 \\ f(x+1)=3x^2+8x+1 \\ f\left(x\right)=3x^2+2x-4 \\ f(x+h)=3{x^2+6xh+3h}^2+2x+2h-4 \end{gathered}$$

Step 1. Given Information

We have to find the value for each function:

$\left(a\right) f\left(0\right) \left(b\right) f\left(1\right) \left(c\right) f(-1) \left(d\right) f(-x) \left(e\right) -f\left(x\right) \left(f\right) f(x+1) \left(g\right) f\left(x\right) \left(h\right) f(x+h)$

The given equation is$f\left(x\right)=3x^2+2x-4$.

Part (a) Step 1. The given function is f ( x ) = 3 x 2 + 2 x - 4

We find the value of $f\left(0\right)$

Part (a) Step 2. Putting x = 0 i the given equation.

$$\begin{gathered} f\left(0\right)=3{\left(0\right)}^2+2\times 0-4 \\ f\left(0\right)=0+0-4 \\ f\left(0\right)=-4 \end{gathered}$$

Part (b) Step 1. We find the value of f ( 1 ) Putting x = 1 in the given question.

$$\begin{gathered} f\left(1\right)=3{\left(1\right)}^2+2\left(1\right)-4 \\ f\left(1\right)=3\times 1+2-4 \\ f\left(1\right)=3+2-4 \\ f\left(1\right)=1 \end{gathered}$$

Part (c) Step 1. We find the value of f ( - 1 ) Putting x = - 1 in the given question.

$$\begin{gathered} f(-1)=3{(-1)}^2+2(-1)-4 \\ f(-1)=3\times 1-2-4 \\ f(-1)=3-2-4 \\ f(-1)=-3 \end{gathered}$$

Part (d) Step 1. We find the value of f ( - x ) Putting x = - x in the given question.

$$\begin{gathered} f(-x)=3{(-x)}^2+2(-x)-4 \\ f(-x)=3x^2-2x-4 \end{gathered}$$

Part (e) Step 1. We find the value of - f ( x ) To find the value we multiply the given equation by - 1

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

Part (f) Step 1. We find the value of f ( x + 1 ) Putting x = x + 1 in the given question.

$f(x+1)=3{(x+1)}^2+2(x+1)-4$

Using the formula ${(a+b)}^2=a^2+2ab+b^2$

$$\begin{gathered} f(x+1)=3(x^2+2x+1)+2x+2-4 \\ f(x+1)=3x^2+6x+3+2x+2-4 \end{gathered}$$

Simplify

$f(x+1)=3x^2+8x+1$

Part (g) Step 1. We find the value of f ( x )

The value of $f\left(x\right)=3x^2+2x-4$

Part (h) Step 1. We find the value of f ( x + h ) Putting x = x + h in the given question.

$f(x+h)=3{(x+h)}^2+2(x+h)-4$

Using the formula ${(a+b)}^2=a^2+2ab+b^2$

$$\begin{gathered} f(x+h)=3{(x^2+2xh+h}^2)+2x+2h-4 \\ f(x+h)=3{x^2+6xh+3h}^2+2x+2h-4 \end{gathered}$$