Chapter 5: Q. 6 (page 348)

In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function.
$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2}$

Short Answer

Expert verified

$(a) (f o g) (x) = \sqrt{3 + 3 x + 3 x^{2}} d o m a i n i s a l l r e a l n n u m b e r s (b) (g o f) (x) = 1 + \sqrt{3 x} + 3 x d o m a i n i s {x | x \geq 0} (c) (f o f) (x) = \sqrt{3 (\sqrt{3 x})} d o m a i n i s {x | x \geq 0} (d) (g o g) (x) = x^{4} + 2 x^{3} + 4 x^{2} + 3 x + 3 d o m a i n i s a l l r e a l n n u m b e r s$

Step by step solution

Sep 1: Given information

we are given with two functions

$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2}$

Part(a) : Step 1: composition of functions

$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2} (f o g) (x) = f (g (x)) = f (1 + x + x^{2}) = \sqrt{3 (1 + x + x^{2})} = \sqrt{3 + 3 x + 3 x^{2}} (f o g) (x) = \sqrt{3 + 3 x + 3 x^{2}}$

domain is all real numbers

Part(b) : Step 1: composition of functions

$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2} (g o f) (x) = g (\sqrt{3 x}) = 1 + \sqrt{3 x} + {(\sqrt{3 x})}^{2} = 1 + \sqrt{3 x} + 3 x (g o f) (x) = 1 + \sqrt{3 x} + 3 x$

Square root cannot be negative .

Domain is {x | x>=0}

Part(c): Step 1: Composition of functions

$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2} (f o f) (x) = f (f (x)) = f (\sqrt{3 x}) = \sqrt{3 (\sqrt{3 x})} (f o f) (x) = \sqrt{3 (\sqrt{3 x})} s q i a r e r o o t o f n e g a t i v e n u m b e r i s i n v a l i d d o m a i n i s {x | x \geq 0}$

Part(d) : step 1: composition of functions

$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2} (g o g) (x) = g (g (x)) = g (1 + x + x^{2}) = 1 + 1 + x + x^{2} + {(1 + x + x^{2})}^{2} (g o g) (x) = 2 + x + x^{2} + x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1 (g o g) (x) = x^{4} + 2 x^{3} + 4 x^{2} + 3 x + 3 D o m a i n i s a l l r e a l n u m b e r s$

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In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function.
$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2}$

Short Answer

Step by step solution

Sep 1: Given information

Part(a) : Step 1: composition of functions

Part(b) : Step 1: composition of functions

Part(c): Step 1: Composition of functions

Part(d) : step 1: composition of functions

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In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function.f(x)=3xg(x)=1+x+x2

Short Answer

Step by step solution

Sep 1: Given information

Part(a) : Step 1: composition of functions

Part(b) : Step 1: composition of functions

Part(c): Step 1: Composition of functions

Part(d) : step 1: composition of functions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 5–7, find f o g, g o f, f o f, and g o g for each pair of functions. State the domain of each composite function.
$f (x) = \sqrt{3 x} g (x) = 1 + x + x^{2}$