10th Maths Exercise 1.5: Composition of Functions Solutions

Exercise 1.5: Composition of Functions

Maths Book back answers and solution for Exercise questions - Mathematics : Function: Composition of Functions: Exercise Problem Questions with Answer

Question 1

Using the functions $f$ and $g$ given below, find $f \circ g$ and $g \circ f$. Check whether $f \circ g = g \circ f$.

1. (i) $f(x) = x - 6$, $g(x) = x^2$
2. (ii) $f(x) = \frac{2}{x}$, $g(x) = 2x^2 – 1$
3. (iii) $f(x) = \frac{x+6}{3}$, $g(x) = 3 - x$
4. (iv) $f(x) = 3 + x$, $g(x) = x - 4$
5. (v) $f(x) = 4x^2 - 1$, $g(x) = 1 + x$

Question 2

Find the value of $k$, such that $f \circ g = g \circ f$

1. (i) $f(x) = 3x + 2$, $g(x) = 6x - k$
2. (ii) $f(x) = 2x - k$, $g(x) = 4x + 5$

Question 3

If $f(x) = 2x - 1$, $g(x) = \frac{x+1}{2}$ show that $f \circ g = g \circ f = x$.

Question 4

1. (i) If $f(x) = x^2 - 1$, $g(x) = x - 2$ find $a$, if $g \circ f(a) = 1$.
2. (ii) Find $k$, if $f(k) = 2k - 1$ and $f \circ f(k) = 5$.

Question 5

Let $A, B, C \subseteq \mathbb{N}$ and a function $f: A \to B$ be defined by $f(x) = 2x + 1$ and $g: B \to C$ be defined by $g(x) = x^2$. Find the range of $f \circ g$ and $g \circ f$.

Question 6

Let $f(x) = x^2 - 1$. Find (i) $f \circ f$ (ii) $f \circ f \circ f$.

Question 7

If $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ are defined by $f(x) = x^5$ and $g(x) = x^4$ then check if $f, g$ are one-one and $f \circ g$ is one-one?

Question 8

Consider the functions $f(x)$, $g(x)$, $h(x)$ as given below. Show that $(f \circ g) \circ h = f \circ (g \circ h)$ in each case.

1. (i) $f(x) = x - 1$, $g(x) = 3x + 1$ and $h(x) = x^2$
2. (ii) $f(x) = x^2$, $g(x) = 2x$ and $h(x) = x + 4$
3. (iii) $f(x) = x - 4$, $g(x) = x^2$ and $h(x) = 3x - 5$

Question 9

Let $f = \{(-1, 3),(0,-1),(2,-9)\}$ be a linear function from $\mathbb{Z}$ into $\mathbb{Z}$. Find $f(x)$.

Question 10

In electrical circuit theory, a circuit $C(t)$ is called a linear circuit if it satisfies the superposition principle given by $C(at_1 + bt_2) = aC(t_1) + bC(t_2)$, where $a,b$ are constants. Show that the circuit $C(t) = 3t$ is linear.

Answers