Question

Find the following for each function:

$f\left(x\right)=\sqrt{x^2-4}$

Part (a):$f\left(2\right)$

Part (b):$f\left(-2\right)$

Part (c):$f\left(-x\right)$

Part (d):$-f\left(x\right)$

Part (e):$f\left(x-2\right)$

Part (f):$f\left(2x\right)$

Short answer

Part (a):$f\left(2\right)=0$

Part (b):$f\left(-2\right)=0$

Part (c):$f\left(-x\right)=\sqrt{x^2-4}$

Part (d):$-f\left(x\right)=-\sqrt{x^2-4}$

Part (e):$f\left(x-2\right)=\sqrt{x^2-4x}$

Part (f):localid="1645859836489" $f\left(2x\right)=2\sqrt{x^2-1}$

Step-by-step solution

Part (a) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\sqrt{x^2-4}$

Substitute $2$in the given equation,

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

Part (b) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\sqrt{x^2-4}$

Substitute $-2$in the given equation,

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

Part (c) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\sqrt{x^2-4}$

Substitute $-x$in the given equation,

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

Part (d) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\sqrt{x^2-4}$

From the question,

$-f\left(x\right)=-\sqrt{x^2-4}$

Part (e) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\sqrt{x^2-4}$

Substitute $x-2$in the given equation,

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

Part (f) Step 1. Given information.

Consider the given function,

$f\left(x\right)=\sqrt{x^2-4}$

Substitute $2x$in the given equation,

localid="1645859830755" $$\begin{gathered} f\left(2x\right)=\sqrt{{\left(2x\right)}^2-4} \\ f\left(2x\right)=\sqrt{4x^2-4} \\ f\left(2x\right)=\sqrt{4\left(x^2-1\right)} \\ f\left(2x\right)=2\sqrt{x^2-1} \end{gathered}$$