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

Previous problemNext problem

1:36 minutes

Problem 61

Textbook Question

Let ƒ(x)=2x-3 and g(x)=-x+3. Find each function value. See Example 5. (g∘ƒ)(0)

Verified step by step guidance

Step 1: Understand the notation \((g \circ f)(x)\), which means the composition of functions \(g\) and \(f\). This is equivalent to \(g(f(x))\).

Step 2: Start by finding \(f(0)\) using the function \(f(x) = 2x - 3\). Substitute \(x = 0\) into \(f(x)\).

Step 3: Calculate \(f(0) = 2(0) - 3\). Simplify this expression to find the value of \(f(0)\).

Step 4: Use the result from Step 3 as the input for the function \(g(x) = -x + 3\). Substitute \(f(0)\) into \(g(x)\) to find \(g(f(0))\).

Step 5: Calculate \(g(f(0)) = -f(0) + 3\). Simplify this expression to find the value of \((g \circ f)(0)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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∘ƒ)(x) means to apply function ƒ first and then apply function g to the result. This process is essential for evaluating composite functions, as it requires substituting the output of one function into another.

Evaluating Functions

Evaluating a function means finding the output value for a given input. For example, to evaluate ƒ(0) for the function ƒ(x)=2x-3, you substitute 0 for x, resulting in ƒ(0)=2(0)-3=-3. This step is crucial in function composition, as you need to compute the value of the inner function before applying the outer function.

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)=2x-3 and g(x)=-x+3 are linear functions, which means their graphs are straight lines. Understanding their properties, such as slope and intercepts, helps in visualizing and solving problems involving these functions.