Chapter 5: Q. 64 (page 257)

If $f (x) = \frac{a x + b}{c x + d}; g (x) = m x,$ then find: -
(a) $f \circ g$
(b) $g \circ f$
(c)Domain of $f \circ g$ and $g \circ f$ .
(d)the conditions for which $f \circ g = g \circ f$ .

Short Answer

Expert verified

(a) $f \circ g (x) = \frac{a m x + b}{c m x + d}$

(b) $g \circ f (x) = \frac{m a x + b}{c x + d}$

(c)Domain of $f \circ g$ is set of real numbers except $x = - \frac{d}{c}$ and domain of $g \circ f$ is set of real numbers.

(d) $f \circ g (x) = g \circ f (x)$ if, $m = 1 .$

Step by step solution

Given Information

Given that $f (x) = \frac{a x + b}{c x + d}; g (x) = m x .$

Part (a) Step 1 Solution

We know that $f \circ g (x) = f (g (x)) .$

Here, $f (x) = \frac{a x + b}{c x + d}; g (x) = m x$

Now,

$f \circ g (x) = f (g (x)) f \circ g (x) = \frac{a (g (x)) + b}{c (g (x)) + d} f \circ g (x) = \frac{a {m x} + b}{c {m x} + d} f \circ g (x) = \frac{a m x + b}{c m x + d} .$

Part (b) Step 1 Solution

We know that $g \circ f (x) = g (f (x)) .$

Here, $f (x) = \frac{a x + b}{c x + d}; g (x) = m x .$

Now,

$g \circ f (x) = m (f (x)) g \circ f (x) = m \{\frac{a x + b}{c x + d}\} g \circ f (x) = = \frac{m a x + b}{c x + d} .$

Part (C) Step 1 Solution

Domain of $f (x) = \frac{a x + b}{c x + d}$ is set of real numbers except than $x = - \frac{d}{c} .$ Domain of $g (x) = m x$ is set of real numbers. So, domain of $f \circ g (x)$ is set of real numbers except then $x = - \frac{d}{c} .$

Now, domain of $g \circ f (x)$ is set of real numbers.

Part (d) Step 1 Solution

Suppose, $f \circ g (x) = g \circ f (x)$

$\frac{a m x + b}{c m x + d} = \frac{m a x + b}{c x + d} c m x + d = c x + d$

compare both sides

$c m = c m = 1 .$

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If $f (x) = \frac{a x + b}{c x + d}; g (x) = m x,$ then find: -
(a) $f \circ g$
(b) $g \circ f$
(c)Domain of $f \circ g$ and $g \circ f$ .
(d)the conditions for which $f \circ g = g \circ f$ .

Short Answer

Step by step solution

Given Information

Part (a) Step 1 Solution

Part (b) Step 1 Solution

Part (C) Step 1 Solution

Part (d) Step 1 Solution

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If f(x)=ax+bcx+d;g(x)=mx,then find: -(a)f∘g(b)g∘f(c)Domain of f∘gand g∘f.(d)the conditions for which f∘g=g∘f.

Short Answer

Step by step solution

Given Information

Part (a) Step 1 Solution

Part (b) Step 1 Solution

Part (C) Step 1 Solution

Part (d) Step 1 Solution

One App. One Place for Learning.

Most popular questions from this chapter

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If $f (x) = \frac{a x + b}{c x + d}; g (x) = m x,$ then find: -
(a) $f \circ g$
(b) $g \circ f$
(c)Domain of $f \circ g$ and $g \circ f$ .
(d)the conditions for which $f \circ g = g \circ f$ .