Practice Questions on Functions and Their Properties with Answers

**Questions on Functions and Their Properties with Answers**

**Q1. The number of distinct real roots of the equation x ^{9}+x^{7}+x^{5}+x^{3}+x+1=0 is**

(a) 1

(b) 3

(c) 5

(d) 9

**Q2. Limit of the function (1- ^{1}/n^{2})^{n} where n → ∞ is equal to**

(a) 1

(b) e^{-1/2}

(c) e^{-2}

(d) e^{-1}

**Q3. Consider the functions f, g : Z→ Z defined by f(n) = 3n + 2 and g(n) = n ^{2} – 5**

(a) Neither f and g is a one-to-one function

(b) Ths function f is one-to-one, but not g

(c) Ths function g is one-to-one, but not f

(d) Both f and g are one-to-one functions

**Q4. Let a,b,c,d be rational number with ad – bc ≠ 0. Then the function f : R\Q → R defined by f(x) = ^{(ax+b)}/_{(cx+d)}**

(a) onto but not one-one

(b) one-one but not onto

(c) neither one-one nor onto

(d) both one-one and onto

**Q5. Let f:(0,∞) → R be the function defined by f(x) = e ^{x}/x^{x}. Then the limit of f(x) where x→∞**

(a) does not exist

(b) exists and is 0

(c) exists and is 1

(d) exists and is e

**Q6. The correct value of limit of the f(x) = ^{x}/_{√(1-cosx)} where x→0**

(a) does not exist

(b) is √2

(c) is -√2

(d) is ^{1}/_{√2}

**Q7. If X and Y are two non-empty finite sets and f:X→Y and g:Y→X are mappings such that g o f : X→X is a surjective (i.e, onto) map, then**

(a) f must be one-to-one

(b) f must be onto

(c) g must be one-to-one

(d) X and Y must have the same number of elements

**Q8. Let P(x) be a non-constant polynomial such that P(n) = P(-n) for all n∈N. Then P'(0)**

(a) Equals 1

(b) Equals 0

(c) Equals -1

(d) Can not be determined from the given data

**Q9. If f(x) = ^{1}/_{(1-x)} , g(x) = f[f(x)] and h(x) = f[g(x)], then what is f(x)g(x)h(x) equal to ?**

(a) -1

(b) 0

(c) 1

(d) 2

**Q10. What is the value of Limit of f(x) = (x ^{y} – y^{x})/(x^{x} – y^{y}) ?**

(a) ^{(1+lny)}/_{(1-lny)}

(b) ^{(1-lny)}/_{(1+lny)}

(c) ^{(-1+lny)}/_{(1+lny)}

(d) ^{(-1-lny)}/_{(1-lny)}

**Q11. If D is the set of all real x such that f(x) = 1 – e ^{(1-x)/x} is positive, then what is D equal to ?**

(a) (-∞,0) ∪ (1,∞)

(b) (-∞,0) ∪ [1,∞)

(c) (1,∞)

(d) (-∞,1)

**Q12. Assertion (A) : e ^{x} cannot be expressed as sum of even and odd functions.**

**Reason (R) : ex is neither even nor odd function**

(a) Both A and R are individually true but R is the correct explanation of A

(b) Both A and R are individually true but R is not the correct explanation of A

(c) A is true but R is false

(d) A is false but R is true

**Q13. Let f:R→R define g:R→R by g(x) = f(x)[f(x) + f(-x)]. Then**

(a) g is even for all f

(b) f is odd for all f

(c) g is even if f is even

(d) g is even if f is odd

**Q14. The limit of function f(x) = (4 ^{x} + 5^{x})^{1/x} where x → ∞ is equal to**

(a) 4

(b) 5

(c) e

(d) 5e

**Q15. Which one of the following function is not well defined ?**

(a) √(1 + sinx)

(b) √(5sec^{2}x – 4)

(c) √(3 – cos^{2}x)

(d) √(x^{4} + x^{2} + ^{1}/_{100})

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*Answer Key*

*Answer Key*

01. | (b) | 06. | (d) | 11. | (a) |

02. | (a) | 07. | (b) | 12. | (d) |

03. | (a) | 08. | (c) | 13. | (d) |

04. | (b) | 09. | (a) | 14. | (b) |

05. | (a) | 10. | (b) | 15. | (d) |