Practice Questions for Algebra - Functions & Graphs

1. CRE 5 - Domain & Range | Algebra - Functions & Graphs

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

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2. CRE 5 - Domain & Range | Algebra - Functions & Graphs

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3. CRE 5 - Domain & Range | Algebra - Functions & Graphs

How many integral values can x take if y = ${[\log_{10} (\frac{6 x - x^{2}}{9})]}^{1 / 2}$

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Answer: 1

Explanation :

Since y is square root of $\log_{10} (\frac{6 x - x^{2}}{9})$

⇒ $\log_{10} (\frac{6 x - x^{2}}{9})$ ≥ 0

⇒ $\frac{6 x - x^{2}}{9}$ ≥ 1

⇒ x² – 6x + 9 ≤ 0

⇒ (x – 3)² ≤ 0

This is only possible when x = 3.

Hence, 1.

4. CRE 5 - Domain & Range | Algebra - Functions & Graphs

Find the domain of the definition of the function y = $\log_{10} (\frac{x - 5}{x^{2} - 10 x + 21})$ + $\sqrt{x - 4}$

A.
x > 7
B.
4 ≤ x < 5
C.
Bothe (a) and (b)
D.
None of these

Answer: Option C

Explanation :

Find the domain of the definition of the function y = $\log_{10} (\frac{x - 5}{x^{2} - 10 x + 21})$ + $\sqrt{x - 4}$

For the function to exist, the argument of the logarithmic function should be positive.

∴ $\frac{x - 3}{x^{2} - 10 x + 21}$ > 0

⇒ $\frac{x - 5}{(x - 3) (x - 7)}$ > 0

⇒ x ∈ (3, 5) ∪ (7, ∞)

Also, (x - 4) ≥ 0 should also be true.

⇒ x ∈ [4, ∞)

∴ x ∈ [4, 5) ∪ (7, ∞)

Hence, option (c).

5. CRE 5 - Domain & Range | Algebra - Functions & Graphs

What is the range of the function f(x) = $\frac{x}{| x |}$ , x ≠ 0?

A.
Any real number
B.
Any integer
C.
{-1, 1}
D.
{-1, 0, 1}

Answer: Option C

Explanation :

As we know |x| = $\{\begin{cases} x & x \geq 0 \\ - x & x < 0 \end{cases}$

∴ f(x) = $\frac{x}{| x |}$ = $\{\begin{cases} \frac{x}{x} & x \geq 0 \\ \frac{x}{- x} & x < 0 \end{cases}$

= $\{\begin{cases} 1 & x \geq 0 \\ - 1 & x < 0 \end{cases}$

∴ Range = {-1, 1}.

Hence, option (c).

6. CRE 5 - Domain & Range | Algebra - Functions & Graphs

Let f(x) = |x| be a function with domain [-6, 4] and let g(x) = |2x + 3|, Then the domain of (fog)(x) is

A.
[-6, 4]
B.
[-7/2, 1/2]
C.
[-7/2, -1/2]
D.
None of these

7. CRE 5 - Domain & Range | Algebra - Functions & Graphs

The domain of the function f(x)= $\frac{\sqrt{(x + 2) (x - 5)}}{x - 7}$ is

A.
x ∈ (-∞, -2) ∪ (5, 7) ∪ (7, ∞)
B.
x ∈ (-∞, -2] ∪ [5, 7) ∪ (7, ∞)
C.
x ∈ (-∞, -2) ∪ (5, ∞)
D.
None of these

Answer: Option B

Explanation :

We are given $\sqrt{(x + 2) (x - 5)}$

∴ (x + 2)(x - 5) ≥ 0

⇒ x ∈ (-∞, -2] ∪ [5, ∞)

Also, the denominator is (x - 7)
∴ x – 7 ≠ 0
⇒ x ≠ 7

∴ Domain is x ∈ (-∞, -2] ∪ [5, 7) ∪ (7, ∞)

Hence, option (b).

8. CRE 5 - Domain & Range | Algebra - Functions & Graphs

If f(x) = log(x²) and g(x) = 2 × log(⁡x), then the domain for which f(x) and g(x) are identical ?

A.
(-∞, ∞)
B.
[0, ∞)
C.
(0, ∞)
D.
None of these

9. CRE 5 - Domain & Range | Algebra - Functions & Graphs

Which of the following pairs are identical?

A.
f(x) = $\sqrt{x^{4}}$ , g(x) = ${(\sqrt{x})}^{4}$
B.
f(x) = $\frac{1}{\sqrt[3]{x^{3}}}$ , g(x) = $\frac{x^{2}}{x^{3}}$
C.
Both (a) and (b)
D.
None of these

Answer: Option B

Explanation :

Option (a):
Domain of f(x) = $\sqrt{x^{4}}$ is R i.e.,x ∈ (-∞,∞)
Domain of g(x) = ${(\sqrt{x})}^{4}$ is R+ ∪ {0} i.e., x ∈ [0, ∞)
∴ $\sqrt{x^{4}}$ ≠ ${(\sqrt{x})}^{4}$

Option (b):
Domain of f(x) = $\frac{1}{\sqrt[3]{x^{3}}}$ is R - {0} and
Domain of g(x) = $\frac{x^{2}}{x^{3}}$ is R – {0}
Also, f(x) = g(x) for x ∈ R – {0}
Hence, option (b).

CRE 5 - Domain & Range | Algebra - Functions & Graphs

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