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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 2 - Functions and GraphsProblema 3
Capitolo 3, Problema 3

Find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x)=3x+8 and g(x) = (x-8)/3

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1
First, find the composition \( f(g(x)) \) by substituting \( g(x) \) into \( f(x) \). This means replacing every \( x \) in \( f(x) = 3x + 8 \) with \( g(x) = \frac{x - 8}{3} \). So, write \( f(g(x)) = 3 \left( \frac{x - 8}{3} \right) + 8 \).
Next, simplify the expression for \( f(g(x)) \) by performing the multiplication and addition inside the function. Multiply 3 by \( \frac{x - 8}{3} \) and then add 8.
Then, find the composition \( g(f(x)) \) by substituting \( f(x) \) into \( g(x) \). Replace every \( x \) in \( g(x) = \frac{x - 8}{3} \) with \( f(x) = 3x + 8 \). So, write \( g(f(x)) = \frac{(3x + 8) - 8}{3} \).
Simplify the expression for \( g(f(x)) \) by performing the subtraction in the numerator and then dividing by 3.
Finally, determine whether \( f \) and \( g \) are inverses by checking if both compositions \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \). If both equal \( x \), then \( f \) and \( g \) are inverse functions.

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Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)) or g(f(x)). It requires substituting the entire expression of one function into the variable of the other, allowing us to combine functions and analyze their combined effect.
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Function Composition

Inverse Functions

Inverse functions reverse the effect of each other, meaning f(g(x)) = x and g(f(x)) = x for all x in the domain. To verify if two functions are inverses, we check if their compositions yield the identity function, which returns the input unchanged.
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Graphing Logarithmic Functions

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions, substituting variables, and solving equations. It is essential for correctly performing function composition and verifying inverse relationships by simplifying the composed functions to check if they equal x.
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Introduction to Algebraic Expressions
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