Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

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40,131 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section provides a comprehensive exploration of functions, starting from basic definitions and criteria to identify a function, determining domains and ranges, and evaluating function values. It also covers operations on functions, including sum, difference, product, and quotient, as well as solving inequalities and expressing results in various notations. Understanding these foundational algebraic concepts is crucial for further study in calculus and applications across multiple disciplines.

Learning Objectives

1

Identify and differentiate between relations and functions, including understanding the role of the domain and range.

2

Determine whether a given relation or equation represents a function using techniques such as mapping, ordered pairs, and the vertical line test.

3

Evaluate functions algebraically, including computing function values, difference quotients, and performing operations on functions (addition, subtraction, multiplication, and division).

4

Find the domain of functions defined by equations by considering restrictions from denominators and even-index radicals.

5

Apply function concepts to real-world problems and interpret the meaning of function outputs in various contexts.

Key Concepts

CONCEPT

DEFINITION

Relation

A correspondence between elements of two sets, where each input is paired with one or more outputs.

Function

A special type of relation in which each element of the domain (input) corresponds to exactly one element in the range (output).

Domain

The set of all possible input values (x-values) for which a function is defined.

Range

The set of all output values (y-values) produced by a function from its domain.

Vertical Line Test

A graphical method to determine if a curve represents a function: if any vertical line intersects the graph at more than one point, it is not a function.

Mapping

A technique of illustrating the relation between two sets by drawing arrows from elements in the domain to corresponding elements in the range.

Difference Quotient

An expression of the form [f(x+h) - f(x)]/h used to measure the average rate of change of a function, and a precursor to the derivative in calculus.

Example Problems

Example 1

The inequality $-1<x<3$ can be written in interval notation as ______.

Example 2

If $x=-2,$ the value of the expression $3 x^{2}-5 x+\frac{1}{x}$ is _______.

Example 3

The domain of the variable in the expression $\frac{x-3}{x+4}$ is _______.

Example 4

Solve the inequality: $3-2 x>5 .$ Graph the solution set. ______

Example 5

If $f$ is a function defined by the equation $y=f(x),$ then $x$ is called the ______ variable and $y$ is the ______ variable.
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Step-by-Step Explanations

QUESTION

Does the equation y = 2x - 5 define y as a function of x?

STEP-BY-STEP ANSWER:

Step 1: Identify the rule given by the equation. Here, multiply the input x by 2 and then subtract 5.
Step 2: Check whether any given x can produce more than one value of y. Since the operations (multiplication and subtraction) are unambiguous for every real number x, there is exactly one output for each input.
Step 3: Conclude that the equation passes the definition of a function because each input x corresponds to only one output y.
Final Answer: Yes, y = 2x - 5 defines y as a function because it assigns exactly one output to each input x.

Determining Whether an Equation Defines a Function

QUESTION

Solve the inequality 3 - 2x ≥ 5 and express the solution in interval notation.

STEP-BY-STEP ANSWER:

Step 1: Start with the given inequality: 3 - 2x ≥ 5.
Step 2: Subtract 3 from both sides to isolate terms involving x: -2x ≥ 2.
Step 3: Divide both sides by -2, remembering to reverse the inequality sign when dividing by a negative number: x ≤ -1.
Step 4: Express the solution in interval notation: (-∞, -1].
Final Answer: The solution is x ≤ -1, which in interval notation is (-∞, -1].

Solving an Inequality and Expressing it in Interval Notation

QUESTION

Determine the domain of f(x) = (3x + 12)/(x - 5).

STEP-BY-STEP ANSWER:

Step 1: Recognize that the domain includes all real numbers except those that make the denominator zero.
Step 2: Set the denominator equal to zero and solve: x - 5 = 0, so x = 5.
Step 3: Exclude x = 5 from the set of all real numbers.
Final Answer: The domain of the function is all real numbers except x = 5, expressed as (-∞, 5) U (5, ∞).

Finding the Domain of a Function

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Common Mistakes

  • Assuming that a relation with repeated output values for different inputs cannot be a function (it is acceptable as long as each input yields only one output).
  • Failing to reverse the inequality sign when dividing by a negative number in inequality problems.
  • Overlooking domain restrictions, for example, not excluding values that cause division by zero or negative values inside even-index radicals.
  • Confusing the domain with the range or misinterpreting the difference between implicit and explicit function definitions.
  • Neglecting proper use of parentheses when evaluating functions or difference quotients, which can lead to algebraic errors.