## Composition of FunctionsGiven the functions $ f(x) = 5x - 4 $ and $ g(x) = x^2 - 5x + 6 $, we are tasked with finding the composition of functions $ (f \circ g)(x) $ and $ (g \circ f)(x) $.### Definitions- **Composition of Functions**: The composition of two functions $ f $ and $ g $ is denoted as $ (f \circ g)(x) $, which means $ f(g(x)) $. Similarly, $ (g \circ f)(x) $ means $ g(f(x)) $.### Calculation1. **Calculate $ (f \circ g)(x) $**: \[ (f \circ g)(x) = f(g(x)) = f(x^2 - 5x + 6) \] Substitute $ g(x) $ into $ f(x) $: \[ f(x^2 - 5x + 6) = 5(x^2 - 5x + 6) - 4 \] Expand the expression: \[ = 5x^2 - 25x + 30 - 4 = 5x^2 - 25x + 26 \]2. **Calculate $ (g \circ f)(x) $**: \[ (g \circ f)(x) = g(f(x)) = g(5x - 4) \] Substitute $ f(x) $ into $ g(x) $: \[ g(5x - 4) = (5x - 4)^2 - 5(5x - 4) + 6 \] Expand the expression: \[ = (25x^2 - 40x + 16) - (25x - 20) + 6 \] Simplify the expression: \[ = 25x^2 - 40x + 16 - 25x + 20 + 6 = 25x^2 - 65x + 42 \]### Conclusion- $ (f \circ g)(x) = 5x^2 - 25x + 26 $

Name: ## Composition of FunctionsGiven the functions $ f(x) = 5x - 4 $ an
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Description: ## Composition of FunctionsGiven the functions $ f(x) = 5x - 4 $ and $ g(x) = x^2 - 5x + 6 $, we are tasked with finding the composition of functions $ (f \circ g)(x) $ and $ (g \circ f)(x) $.### Definitions- **Composition of Functions**: Th

givenf(x)=5x-4and g(x)=x2-5x+6fog(x)=f{g(x)}=f(x2-5x+6)=5(x2-5x+6)-4=5x2-25x+26 answer

Answered: Let f(x) = 5x - 4 and g(x) = x? – 5x +…

Math

Algebra

Let f(x) = 5x - 4 and g(x) = x? – 5x + 6. Then %3D %3D (f o g)(x) = %3D (go f)(x) = %3D

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.2: Domain And Range

Problem 2SE: How do we determine the domain of a function defined by an equation?

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Question

## Composition of Functions

Given the functions $ f(x) = 5x - 4 $ and $ g(x) = x^2 - 5x + 6 $, we are tasked with finding the composition of functions $ (f \circ g)(x) $ and $ (g \circ f)(x) $.

### Definitions

- **Composition of Functions**: The composition of two functions $ f $ and $ g $ is denoted as $ (f \circ g)(x) $, which means $ f(g(x)) $. Similarly, $ (g \circ f)(x) $ means $ g(f(x)) $.

### Calculation

1. **Calculate $ (f \circ g)(x) $**:
\[
(f \circ g)(x) = f(g(x)) = f(x^2 - 5x + 6)
\]
Substitute $ g(x) $ into $ f(x) $:
\[
f(x^2 - 5x + 6) = 5(x^2 - 5x + 6) - 4
\]
Expand the expression:
\[
= 5x^2 - 25x + 30 - 4 = 5x^2 - 25x + 26
\]

2. **Calculate $ (g \circ f)(x) $**:
\[
(g \circ f)(x) = g(f(x)) = g(5x - 4)
\]
Substitute $ f(x) $ into $ g(x) $:
\[
g(5x - 4) = (5x - 4)^2 - 5(5x - 4) + 6
\]
Expand the expression:
\[
= (25x^2 - 40x + 16) - (25x - 20) + 6
\]
Simplify the expression:
\[
= 25x^2 - 40x + 16 - 25x + 20 + 6 = 25x^2 - 65x + 42
\]

### Conclusion

- $ (f \circ g)(x) = 5x^2 - 25x + 26 $

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