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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 109
Chapter 5, Problem 109

Use the properties of inverses to determine whether ƒ and g are inverses. ƒ(x) = log￬4 (x+3), g(x) = 4x + 3

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Recall that two functions \( f \) and \( g \) are inverses if and only if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of the compositions.
Start by finding the composition \( f(g(x)) \). Substitute \( g(x) = 4^x + 3 \) into \( f(x) = \log_4 (x + 3) \), so \( f(g(x)) = \log_4 ((4^x + 3) + 3) \).
Simplify the expression inside the logarithm: \( (4^x + 3) + 3 = 4^x + 6 \). So, \( f(g(x)) = \log_4 (4^x + 6) \).
Next, find the composition \( g(f(x)) \). Substitute \( f(x) = \log_4 (x + 3) \) into \( g(x) = 4^x + 3 \), so \( g(f(x)) = 4^{\log_4 (x + 3)} + 3 \).
Simplify \( 4^{\log_4 (x + 3)} \) using the property that \( a^{\log_a b} = b \), so \( g(f(x)) = (x + 3) + 3 = x + 6 \). Compare both compositions to \( x \) to determine if \( f \) and \( g \) are inverses.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

Inverse functions reverse each other's operations, meaning if f and g are inverses, then f(g(x)) = x and g(f(x)) = x for all x in their domains. Understanding this relationship is essential to verify if two functions are inverses.
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Graphing Logarithmic Functions

Logarithmic and Exponential Functions

Logarithmic functions are the inverses of exponential functions with the same base. For example, log base 4 and 4 raised to a power undo each other, which is key to analyzing the given functions f(x) = log₄(x+3) and g(x) = 4^x + 3.
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Graphs of Logarithmic Functions

Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)) or g(f(x)). To check if two functions are inverses, you compose them in both orders and verify if the result simplifies to the identity function x.
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Function Composition
Related Practice
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