Textbook Question
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = −1, passing through (−4, − 1/4)
Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 20
Determine whether each equation defines y as a function of x.
Verified step by step guidance1
Recall that a relation defines y as a function of x if for every x-value in the domain, there is exactly one corresponding y-value.
Look at the given equation: \(y = - \sqrt{x} + 4\). The square root function \(\sqrt{x}\) is defined only for \(x \geq 0\).
Since the square root function outputs only the principal (non-negative) root, \(\sqrt{x}\) is always non-negative, so \(-\sqrt{x}\) is always non-positive.
For each \(x \geq 0\), the expression \(-\sqrt{x} + 4\) produces exactly one value of \(y\), because the square root function is single-valued and the operations are well-defined.
Therefore, the equation defines \(y\) as a function of \(x\) because each \(x\) in the domain corresponds to exactly one \(y\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of a Function
A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, check if for every x there is only one y. If multiple y-values exist for a single x, it is not a function.
Recommended video:
5:57
Graphs of Common Functions
Square Root Function and Domain
The square root function, √x, is defined only for x ≥ 0 because the square root of a negative number is not real. When analyzing y = -√x + 4, the domain is restricted to x ≥ 0, ensuring the expression under the root is valid.
Recommended video:
02:20Imaginary Roots with the Square Root Property
Evaluating Function Output Uniqueness
To verify if y = -√x + 4 defines y as a function of x, substitute values of x and observe the output y. Since the square root yields a single non-negative value and the equation applies a negative sign and addition, each x produces exactly one y, confirming it is a function.
Recommended video:
4:26Evaluating Composed Functions
Related Practice
Textbook Question
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−1, 5) and is perpendicular to the line whose equation is x = 6.
Textbook Question
Use the graph of y = f(x) to graph each function g. g(x) = f(x+1)
Textbook Question
The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = (x+2)³
Textbook Question
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−2, 6) and is perpendicular to the line whose equation is x = -4.
Textbook Question
Determine whether each equation defines y as a function of x. y = √x +4