Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

0. Review of College Algebra

Functions

Trigonometry

0. Review of College Algebra

Functions

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1:24 minutes

Problem 49

Textbook Question

Let f(x) = -3x + 4 and g(x) = -x² + 4x + 1. Find each of the following. Simplify if necessary. See Example 6.ƒ(⅓)

Verified step by step guidance

Identify the function f(x) = -3x + 4.

Substitute x = \frac{1}{3} into the function f(x).

Calculate f(\frac{1}{3}) by replacing x with \frac{1}{3} in the expression: f(\frac{1}{3}) = -3(\frac{1}{3}) + 4.

Simplify the expression: -3(\frac{1}{3}) + 4.

Combine the terms to find the simplified result.

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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, to evaluate f(x) at x = ⅓, you replace x in the function f(x) = -3x + 4 with ⅓, resulting in f(⅓) = -3(⅓) + 4. This process is fundamental in understanding how functions behave at particular points.

Linear Functions

A linear function is a polynomial function of degree one, represented 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 graphs and behaviors.

Simplification of Expressions

Simplification of expressions involves reducing mathematical expressions to their simplest form, making them easier to work with. This can include combining like terms, factoring, or performing arithmetic operations. In the context of evaluating f(⅓), simplifying the result after substitution ensures clarity and accuracy in the final answer.