Question

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

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

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

Short answer

The value of

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

Step-by-step solution

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

Part (a) Step 1. We find the value of  f(0)Putting x=0 in the given question.

$$\begin{gathered} f\left(0\right)=-2{\left(0\right)}^2+0-1 \\ f\left(0\right)=-1 \end{gathered}$$

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

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

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

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

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

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

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

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

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

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

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

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

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

The value of$f\left(x\right)=-2x^2+x-1$

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

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

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

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