Chapter 2, Problem 4RE

In Problems 3-5, find the following for each function.

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

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

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

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

(e) $f\left(x-\text{ 2}\right)$

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

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

To determine

To find: The value of the provided function by substitution method.

Answer to Problem 4RE

Solution:

a. $f\left(2\right)=0$

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=\sqrt{x^2-4}$.

Calculation:

a. For $f\left(2\right)$, substitute $x$ as 2. That is, $f\left(2\right)=\sqrt{{\left(2\right)}^2-4}=0$.

To determine

To find: The value of the provided function by substitution method.

Answer to Problem 4RE

Solution:

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

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=\sqrt{x^2-4}$.

Calculation:

b. For $f\left(-2\right)$, substitute $x$ as $-2$. That is, $f\left(-2\right)=\sqrt{{\left(-2\right)}^2-4}=0$.

To determine

To find: The value of the provided function by substitution method.

Answer to Problem 4RE

Solution:

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

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=\sqrt{x^2-4}$.

Calculation:

b. For $f\left(-2\right)$, substitute $x$ as $-2$. That is, $f\left(-2\right)=\sqrt{{\left(-2\right)}^2-4}=0$.

To determine

To find: The value of the provided function by substitution method.

Answer to Problem 4RE

Solution:

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

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=\sqrt{x^2-4}$.

Calculation:

c. For $f\left(-x\right)$, substitute $x$ as $-x$. That is, $f\left(-x\right)=\sqrt{{\left(-x\right)}^2-4}=\sqrt{x^2-4}$.

To determine

To find: The value of the provided function by substitution method.

Answer to Problem 4RE

Solution:

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

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=\sqrt{x^2-4}$.

Calculation:

d. For $-f\left(x\right)$, multiply by $-1$. That is, $-f\left(x\right)=-\left(\sqrt{x^2-4}\right)=\sqrt{x^2-4}$.

To determine

To find: The value of the provided function by substitution method.

Answer to Problem 4RE

Solution:

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

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=\sqrt{x^2-4}$.

Calculation:

e. For $f\left(x-2\right)$, substitute $x$ as $x-2$. That is, $f\left(x-2\right)=\sqrt{{\left(x-2\right)}^2-4}=\sqrt{x^2+4-4x-4}=\sqrt{x^2-4x}=\sqrt{x\left(x-4\right)}$.

To determine

To find: The value of the provided function by substitution method.

Answer to Problem 4RE

Solution:

f. $f\left(2x\right)=2\sqrt{x^2-1}$

Explanation of Solution

Given:

It is provided in the problem that the function $f\left(x\right)=\sqrt{x^2-4}$.

Calculation:

f. For $f\left(2x\right)$, substitute $x$ as $2x$. That is, $f\left(2x\right)=\sqrt{{\left(2x\right)}^2-4}=\sqrt{4x^2-4}=\sqrt{4\left(x^2-1\right)}=2\sqrt{x^2-1}$.