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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2:34 minutes

Problem 59

Textbook Question

In Exercises 51–66, find a. (fog) (2)b. (go f) (2) f(x) = 4-x, g(x) = 2x² +x+5

Verified step by step guidance

Step 1: Understand the composition of functions. (f o g)(x) means f(g(x)) and (g o f)(x) means g(f(x)).

Step 2: Calculate (f o g)(x). Substitute g(x) into f(x): f(g(x)) = f(2x^2 + x + 5).

Step 3: Substitute g(x) into f(x): f(2x^2 + x + 5) = 4 - (2x^2 + x + 5).

Step 4: Simplify the expression: 4 - (2x^2 + x + 5) = 4 - 2x^2 - x - 5.

Step 5: Calculate (f o g)(2) by substituting x = 2 into the simplified expression.

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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. If f(x) and g(x) are two functions, the composition (fog)(x) means applying g first and then f to the result, expressed as f(g(x)). Understanding this concept is crucial for solving the given problem, as it requires evaluating the functions in a specific order.

Evaluating Functions

Evaluating functions means substituting a specific input value into the function to find the output. For example, if f(x) = 4 - x, to evaluate f(2), you would substitute 2 for x, resulting in f(2) = 4 - 2 = 2. This skill is essential for calculating the values of (fog)(2) and (go f)(2) in the exercise.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c. In this problem, g(x) = 2x² + x + 5 is a quadratic function. Understanding its properties, such as its shape (a parabola) and how to manipulate it, is important for performing the composition with f(x) and evaluating the results.