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

Intro to Functions & Their Graphs

College Algebra

3. Functions

Intro to Functions & Their Graphs

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1:10 minutes

Problem 63

Textbook Question

Let ƒ(x)=-3x+4 and g(x)=-x^2+4x+1. Find each of the following. Simplify if necessary. See Example 6. ƒ(x+2)

Verified step by step guidance

Substitute (x+2) for x in the function ƒ(x) = -3x + 4. This means wherever you see an x in the function, replace it with (x+2).

Rewrite the function with the substitution: ƒ(x+2) = -3(x+2) + 4.

Distribute the -3 inside the parentheses: -3(x+2) = -3x - 6.

Combine the distributed terms with the constant in the function: -3x - 6 + 4.

Simplify the expression by combining like terms: -3x - 2.

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Key Concepts

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

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. In this case, evaluating ƒ(x+2) means replacing every instance of x in the function ƒ(x) with (x+2). This process is essential for finding the value of the function at a new input.

Linear Functions

A linear function is a polynomial function of degree one, which can be expressed in the form ƒ(x) = mx + b, where m is the slope and b is the y-intercept. The function ƒ(x) = -3x + 4 is linear, indicating that its graph is a straight line. Understanding the properties of linear functions helps in analyzing their behavior and transformations.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form by combining like terms, factoring, or applying arithmetic operations. After evaluating ƒ(x+2), it may be necessary to simplify the resulting expression to make it easier to interpret or use in further calculations. Mastery of simplification techniques is crucial in algebra.