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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0:42 minutes

Problem 59

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(p)

Verified step by step guidance

Identify the function ƒ(x) which is given as -3x + 4.

Replace x with p in the function ƒ(x). This means wherever you see x, substitute it with p.

Write the new function expression by substituting x with p: ƒ(p) = -3p + 4.

This expression, -3p + 4, is the simplified form of ƒ(p).

There are no further simplifications needed, so ƒ(p) = -3p + 4 is the final expression.

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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 value into a function to determine its output. For example, if you have a function f(x) and you want to find f(p), you replace x with p in the function's expression. This process is fundamental in algebra as it allows you to compute specific outputs based on given inputs.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the given function f(x) = -3x + 4, the slope is -3, indicating a decrease in value as x increases, while the y-intercept is 4, showing where the line crosses the y-axis. Understanding linear functions is crucial for analyzing their behavior and graphing them.

Quadratic Functions

A quadratic function is a polynomial function of degree two, generally represented as g(x) = ax^2 + bx + c. In the function g(x) = -x^2 + 4x + 1, the leading coefficient is negative, indicating that the parabola opens downward. Quadratic functions are important in algebra for modeling various real-world scenarios and understanding their properties, such as vertex and intercepts.