Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 48

Chapter 3, Problem 48

Determine whether each relation defines y as a function of x. Give the domain and range. y=-7/(x-5)

Verified step by step guidance

Identify the given relation: $y = \frac{-7}{x - 5}$. This is a rational function where $y$ is expressed in terms of $x$.

Determine if $y$ is a function of $x$: For each value of $x$ (except where the expression is undefined), there is exactly one corresponding value of $y$. Since the expression is a fraction with $x$ in the denominator, check where the denominator is zero.

Find the domain by setting the denominator not equal to zero: Solve $x - 5 \neq 0$, which gives $x \neq 5$. So, the domain is all real numbers except $x = 5$.

Since for each $x$ in the domain there is exactly one $y$, the relation defines $y$ as a function of $x$.

Determine the range: Consider the values $y$ can take. Since $y = \frac{-7}{x - 5}$, $y$ can be any real number except possibly some value that makes the expression undefined or impossible. Analyze the behavior of the function to describe the range.

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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 corresponds to exactly one output y. To determine if y is a function of x, check that for every x-value, there is only one y-value. If any x maps to multiple y-values, the relation is not a function.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For rational functions like y = -7/(x-5), the domain excludes values that make the denominator zero, since division by zero is undefined.

Range of a Function

The range is the set of all possible output values (y-values) that the function can produce. To find the range, analyze the behavior of the function and determine which y-values are attainable given the domain restrictions.

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