Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

3. Functions

Function Operations

College Algebra

3. Functions

Function Operations

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1:29 minutes

Problem 64

Textbook Question

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (g∘g)(-2)

Verified step by step guidance

Identify the composition of functions: \((g \circ g)(x) = g(g(x))\).

Substitute \(g(x) = -x + 3\) into the composition: \(g(g(x)) = g(-x + 3)\).

Apply the function \(g\) to \(-x + 3\): \(g(-x + 3) = -(-x + 3) + 3\).

Simplify the expression: \(-(-x + 3) + 3 = x - 3 + 3\).

Evaluate the simplified expression at \(x = -2\).

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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 combining two functions to create a new function. The notation (g∘g)(x) means to apply the function g to itself. This requires first finding g(x) and then substituting that result back into g. Understanding this concept is crucial for evaluating composite functions correctly.

Evaluating Functions

Evaluating a function means substituting a specific input value into the function's formula to find the corresponding output. For example, to evaluate g(-2), you replace x in g(x) with -2. This process is fundamental in finding function values and is necessary for solving problems involving function composition.

Linear Functions

Linear functions are mathematical expressions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. Both ƒ(x) and g(x) in the question are linear functions. Understanding their properties, such as how to graph them and their behavior, is essential for manipulating and composing these functions effectively.