Relations And Functions Test

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Relations And Functions Test - 1

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Question 1
1 / -0

Which of the following functions is an even function?

A
f(x) = x(x² – x⁴)

B
f(x) = sin(5x + x³)

C
f(x) = x² – 3x + 4

D
f(x) = x² + |x|

Solution

For a function to be an even function, f(–x) = f(x) for all x.

Consider (1): f(x) = x(x² – x⁴) = x³ – x⁵

f(–x) = (–x)³ – (–x)⁵ = –x³ + x⁵ = –(x³ – x⁵), f(–x) ≠ f(x)

Thus, f(x) is not an even function. It is an odd function as f(–x) = –f(x).

Consider (2): f(x) = sin(5x + x³)

f(–x) = sin(–5x + (–x)³) = sin(–5x – x³) = sin(–(5x + x³)) = –sin(5x + x³)

f(–x) ≠ f(x)

This function is not an even function. It is an odd function.

Consider (3): f(x) = x² – 3x + 4; f(–x) = x² + 3x + 4, f(–x) ≠ f(x)

Also, f(–x) ≠ –f(x)

This function is neither an even function nor an odd function.

Consider (4): f(x) = x² + |x|

f(–x) = x² + |–x| = x² + |x| = f(x)

This function is an even function.
Question 2
1 / -0

The domain of the function f = {(1, 3), (3, 5), (2, 6)} is

A
1, 3 and 2

B
{1, 3, 2}

C
{3, 5, 6}

D
3, 5 and 6

Solution

Domain of f is a set of those points on which f is defined.
Question 3
1 / -0

The range of the function f(x) = |x - 1| is

A
(- ∞, ∞)

B
(0, ∞)

C
[0, ∞)

D
(- ∞, 0)

Solution

Since |x| assumes every value in [0, ∞), so |x - 1| also takes every value in [0, ∞). Hence, range of f(x) = |x - 1| is [0, ∞).