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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1:06 minutes

Problem 26

Textbook Question

Determine whether each equation defines y as a function of x. x = (1/3)(y^2)

Verified step by step guidance

Step 1: Recall the definition of a function. A relation is a function if each input (x-value) corresponds to exactly one output (y-value).

Step 2: Consider the given equation: \( x = \frac{1}{3}y^2 \). We need to determine if for each x-value, there is only one corresponding y-value.

Step 3: Solve the equation for \( y \) in terms of \( x \). Multiply both sides by 3 to get \( 3x = y^2 \).

Step 4: Take the square root of both sides to solve for \( y \). This gives \( y = \pm \sqrt{3x} \).

Step 5: Analyze the result. Since \( y = \pm \sqrt{3x} \) gives two possible values of \( y \) for each positive \( x \), the equation does not define \( y \) as a function of \( x \).

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Key Concepts

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

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines one variable as a function of another.

Vertical Line Test

The vertical line test is a visual way to determine if a curve is a function. If any vertical line intersects the graph of the relation at more than one point, then the relation does not define y as a function of x. This test helps in understanding the graphical representation of functions and their properties.

Solving for y

To determine if an equation defines y as a function of x, it is often necessary to solve the equation for y. This involves isolating y on one side of the equation. If the resulting expression for y can be expressed as a single output for each input x, then y is a function of x; otherwise, it is not.