Chapter 5: Q. 63 (page 257)

If $f (x) = a x + b$ and $g (x) = c x + d$ , then find: -
(a) $f \circ g$
(b) $g \circ f$
(c)the 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) $f \circ g (x) = a c x + a d + b$

(b) $g \circ f (x) = a c x + b c + d$

(c)Domain of $f \circ g$ and $g \circ f$ are same, that is set of real numbers.

(d) $f \circ g$ and $g \circ f$ are same if $a d + b = b c + d .$

Step by step solution

Step 1. Given Information

Given that $f (x) = a x + b$ and $g (x) = c x + d .$

Part (a) step 1. solution

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

Here, $f (x) = a x + b$ , $g (x) = c x + d .$

then,

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

Part (b) Step 1. Solution

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

Here, $f (x) = a x + b$ , $g (x) = c x + d$

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

Part (c) Step 1. Solution

Since the domain of $f (x)$ and $g (x)$ are set of real number so domain of $f \circ g$ and $g \circ f$ are also set of real number.

Part (d) Step 1. Solution

We know that $f \circ g (x) = a c x + a d + b$ and $g \circ f (x) = a c x + b c + d$ .

Now,

localid="1646279876516" $f \circ g (x) = g \circ f (x) a c x + a d + b = a c x + b c + d$

Compare coefficients on both sides.

$a d + b = b c + d$ .

So, $f \circ g = g \circ f$

if, $a d + b = b c + d .$

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If $f (x) = a x + b$ and $g (x) = c x + d$ , then find: -
(a) $f \circ g$
(b) $g \circ f$
(c)the 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

Step 1. 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+band g(x)=cx+d, then find: -(a)f∘g(b)g∘f(c)the domain of f∘gand g∘f(d)the conditions for whichf∘g=g∘f

Short Answer

Step by step solution

Step 1. 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.

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If $f (x) = a x + b$ and $g (x) = c x + d$ , then find: -
(a) $f \circ g$
(b) $g \circ f$
(c)the domain of $f \circ g$ and $g \circ f$
(d)the conditions for which $f \circ g = g \circ f$