CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

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Answer the next 5 questions based on the information given below.

Determine which of the following functions are Even or odd or neither.

Type 1: if the function is even.
Type 2: if the function is odd.
Type 3: if the function is neither even nor odd.

1. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

f(x) = 2x².

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7 8 9 4 5 6 1 2 3 0 . -

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2. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

f(x) = 3x³.

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7 8 9 4 5 6 1 2 3 0 . -

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3. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

f(x) = 2x² + 3x³.

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7 8 9 4 5 6 1 2 3 0 . -

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4. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

f(x) = 2x² + |x|.

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7 8 9 4 5 6 1 2 3 0 . -

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5. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

f(x) = 3x³ + x.

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7 8 9 4 5 6 1 2 3 0 . -

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6. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

If the function f(x) satisfies the relation f(x + y) + f(x - y) = 2f(x)f(y), such that x and y belong to the set of real numbers, and f(0) ≠ 0, then f(x) is

(a)
Even
(b)
Odd
(c)
Neither even nor odd
(d)
Both even as well as odd

7. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

If f(x) is an even function, then the graph y=f(x) will be symmetrical about

(a)
x - axis
(b)
y - axis
(c)
Both the axes
(d)
None of these

8. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

If f(x) = 3xⁿ and g(x) = 5x⁽ⁿ⁺¹⁾, where n is a natural number. Find f(x).g(x) is which kind of function?

(a)
Even function
(b)
Odd function
(c)
Neither Even nor Odd function
(d)
Cannot be determined

9. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

Let f(x) = |x - 3| + |x - 4| + |x - 5| and g(x) = [f(x)]². Then

(a)
One of f(x) or g(x) is an even function
(b)
One of f(x) or g(x) is an odd function
(c)
f(x) and g(x) are both either even or odd
(d)
None of these

10. CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

Which of the following functions is an odd function?

(a)
$5^{| x^{x} |}$
(b)
$5^{| x |^{| x |}}$
(c)
Both (a) and (b)
(d)
Neither (a) nor (b)

Answer: Option D

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Explanation :

Consider f(x) = |x^x|
f(2) = 2² = 4, and
f(-2) = -2^-2 = -1/4
∵ f(2) ≠ f(-2) hence, f(x) is neither even nor odd.
⇒ $5^{| x^{x} |}$ is neither even nor odd

Now, consider g(x) = |x|^|x|
g(-x) = |-x|^|-x| = |x|^|x|
∴ g(x) is an even function.
⇒ $5^{| x |^{| x |}}$ is also an even function

Hence, option (d).

CRE 2 - Even / Odd Functions | Algebra - Functions & Graphs

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