Question

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

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

$f\left(x\right)=\sqrt{x^2+x}$

Short answer

The value of

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

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$f\left(x\right)=\sqrt{x^2+x}$

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

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

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

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

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

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

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

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

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

$-f\left(x\right)=-\sqrt{x^2+x}$

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

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

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

localid="1645962442431" $$\begin{gathered} f(x+1)=\sqrt{x^2+2x+1+x+1} \\ f(x+1)=\sqrt{x^2+3x+2} \end{gathered}$$

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

The value of$f\left(x\right)=\sqrt{x^2+x}$

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

$f(x+h)=\sqrt{{(x+h)}^2+x+h}$

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

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