Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 64

Chapter 3, Problem 64

Let f(x) = 2x - 5 g(x) = 4x - 1 h(x) = x² + x + 2. Evaluate the indicated function without finding an equation for the function. f(g[h (1)])

Verified step by step guidance

First, identify the innermost function to evaluate, which is \( h(1) \). Substitute \( x = 1 \) into \( h(x) = x^2 + x + 2 \) to find \( h(1) \).

Next, take the result from \( h(1) \) and substitute it into \( g(x) = 4x - 1 \) to find \( g(h(1)) \).

Then, take the result from \( g(h(1)) \) and substitute it into \( f(x) = 2x - 5 \) to find \( f(g(h(1))) \).

Remember to perform each substitution step carefully, simplifying the expressions at each stage before moving to the next function.

By following these steps, you will evaluate \( f(g(h(1))) \) without needing to find the composite function's explicit formula.

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Key Concepts

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

Function Composition

Function composition involves applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). In this problem, you evaluate functions inside out, starting with the innermost function h(1), then applying g to that result, and finally f to the output of g.

Evaluating Functions at a Given Input

Evaluating a function means substituting a specific value for the variable and simplifying. For example, to find h(1), replace x with 1 in h(x) = x² + x + 2 and calculate the result. This step-by-step substitution is essential for nested function evaluation.

Order of Operations in Nested Functions

When dealing with nested functions like f(g(h(1))), you must follow the order from the innermost function outward. First evaluate h(1), then use that result as input for g, and finally apply f to the output of g. This ensures accurate and systematic calculation.

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