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 Composition

College Algebra

3. Functions

Function Composition

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2:08 minutes

Problem 73

Textbook Question

In Exercises 67-74, finda. (fog) (x)b. the domain of f o g. f(x) = x² + 4, g(x) = √(1 − x)

Verified step by step guidance

Step 1: Understand the composition of functions. The composition (f \circ g)(x) means applying g first and then applying f to the result.

Step 2: Substitute g(x) into f(x). Since f(x) = x^2 + 4 and g(x) = \sqrt{1 - x}, substitute g(x) into f to get f(g(x)) = (\sqrt{1 - x})^2 + 4.

Step 3: Simplify the expression. Simplifying (\sqrt{1 - x})^2 gives 1 - x, so f(g(x)) = 1 - x + 4.

Step 4: Determine the domain of g(x). The function g(x) = \sqrt{1 - x} is defined when the expression under the square root is non-negative, so 1 - x \geq 0, which simplifies to x \leq 1.

Step 5: Determine the domain of f(g(x)). Since f(g(x)) = 1 - x + 4 is defined for all real numbers, the domain of f \circ g is the same as the domain of g(x), which is x \leq 1.

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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, where the output of one function becomes the input of another. In this case, (f o g)(x) means applying g first and then applying f to the result of g. Understanding how to correctly substitute and evaluate these functions is crucial for solving the problem.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the composition of functions, the domain of f o g is determined by the domain of g and the values that g outputs that are also valid inputs for f. This requires analyzing both functions to ensure all inputs are permissible.

Square Root Function

The square root function, denoted as g(x) = √(1 - x), is defined only for non-negative values under the square root. This means that the expression 1 - x must be greater than or equal to zero, which imposes restrictions on the domain of g. Understanding these restrictions is essential for determining the overall domain of the composite function f o g.