Chapter 2: Q. 4 (page 124)

Find the following for each function:
$f (x) = \sqrt{x^{2} - 4}$
Part (a): $f (2)$
Part (b): $f (- 2)$
Part (c): $f (- x)$
Part (d): $- f (x)$
Part (e): $f (x - 2)$
Part (f): $f (2 x)$

Short Answer

Expert verified

Part (a): $f (2) = 0$

Part (b): $f (- 2) = 0$

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

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

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

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

Step by step solution

Part (a) Step 1. Given information.

Consider the given function,

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

Substitute $2$ in the given equation,

$f (2) = \sqrt{{(2)}^{2} - 4} f (2) = \sqrt{4 - 4} f (2) = 0$

Part (b) Step 1. Given information.

Consider the given function,

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

Substitute $- 2$ in the given equation,

$f (- 2) = \sqrt{{(- 2)}^{2} - 4} f (- 2) = \sqrt{4 - 4} f (- 2) = 0$

Part (c) Step 1. Given information.

Consider the given function,

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

Substitute $- x$ in the given equation,

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

Part (d) Step 1. Given information.

Consider the given function,

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

From the question,

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

Part (e) Step 1. Given information.

Consider the given function,

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

Substitute $x - 2$ in the given equation,

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

Part (f) Step 1. Given information.

Consider the given function,

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

Substitute $2 x$ in the given equation,

localid="1645859830755" $f (2 x) = \sqrt{{(2 x)}^{2} - 4} f (2 x) = \sqrt{4 x^{2} - 4} f (2 x) = \sqrt{4 (x^{2} - 1)} f (2 x) = 2 \sqrt{x^{2} - 1}$

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Find the following for each function:
$f (x) = \sqrt{x^{2} - 4}$
Part (a): $f (2)$
Part (b): $f (- 2)$
Part (c): $f (- x)$
Part (d): $- f (x)$
Part (e): $f (x - 2)$
Part (f): $f (2 x)$

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (b) Step 1. Given information.

Part (c) Step 1. Given information.

Part (d) Step 1. Given information.

Part (e) Step 1. Given information.

Part (f) Step 1. Given information.

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Find the following for each function:fx=x2-4Part (a):f2Part (b):f-2Part (c):f-xPart (d):-fxPart (e):fx-2Part (f):f2x

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (b) Step 1. Given information.

Part (c) Step 1. Given information.

Part (d) Step 1. Given information.

Part (e) Step 1. Given information.

Part (f) Step 1. Given information.

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Find the following for each function:
$f (x) = \sqrt{x^{2} - 4}$
Part (a): $f (2)$
Part (b): $f (- 2)$
Part (c): $f (- x)$
Part (d): $- f (x)$
Part (e): $f (x - 2)$
Part (f): $f (2 x)$