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Chapter 0: Problem 45

Find the domain of each function given below. $$ f(x)=\frac{3 x-1}{7-2 x} $$

Short Answer

Expert verified
The domain is all real numbers except \( x = \frac{7}{2} \).

Step by step solution

01

Identify Domain Restrictions

First, observe the function \( f(x) = \frac{3x - 1}{7 - 2x} \). The denominator \( 7 - 2x \) must not be zero to ensure the function is defined. Thus, find the values of \( x \) that make the denominator zero and exclude them from the domain.
02

Solve for Restrictions

Set the denominator equal to zero and solve for \( x \): \( 7 - 2x = 0 \). Rearrange to find \( 2x = 7 \), thus \( x = \frac{7}{2} \). This is the value that makes the denominator zero, so \( x = \frac{7}{2} \) must be excluded from the domain.
03

State the Domain

The domain of the function \( f(x) \) is all real numbers except \( x = \frac{7}{2} \). In interval notation, the domain is: \( (-\infty, \frac{7}{2}) \cup (\frac{7}{2}, \infty) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Rational functions are expressions that consist of a ratio of two polynomials. The general form is \( R(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
  • The function is undefined where the denominator \( Q(x) \) equals zero.
  • The degree of \( P(x) \) can be less than, equal to, or greater than the degree of \( Q(x) \), affecting the function's asymptotic behavior.

Consider the example \( f(x) = \frac{3x-1}{7-2x} \). Here, \( 3x-1 \) is the numerator and \( 7-2x \) is the denominator.
The key thing to remember with rational functions is checking for points where the denominator becomes zero, as these are typically where the function is not defined.
Domain Restrictions
Domain restrictions are crucial for determining where a function is defined. For rational functions, any values of \( x \) that make the denominator zero must be excluded from the domain.

To find domain restrictions, follow these steps:
  • Identify the denominator of the rational function.
  • Set the denominator equal to zero and solve for \( x \).
  • Exclude these \( x \) values from the domain.

For \( f(x) = \frac{3x-1}{7-2x} \), setting the denominator to zero gives \( 7-2x=0 \). Solving this equation, we find \( x = \frac{7}{2} \) as a restriction since \( x = \frac{7}{2} \) would make the denominator, and thus the function undefined.
Interval Notation
Interval notation is a way to describe the domain of a function, particularly useful for specifying which parts of the number line are included or excluded.

Intervals are denoted using parentheses \(( )\) and brackets \([ ]\):
  • Use parentheses \(( )\) to indicate that an endpoint is not included.
  • Use brackets \([ ]\) to include an endpoint.

For the function \( f(x) = \frac{3x-1}{7-2x} \), the domain is all real numbers except \( x = \frac{7}{2} \). Hence, we express it in interval notation as: \((-\infty, \frac{7}{2}) \cup (\frac{7}{2}, \infty)\).
Here, \( \cup \) signifies a union of intervals, joining the segments on either side of the excluded value.

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Most popular questions from this chapter

The Technology Connection heading indicates exercises designed to provide practice using a graphing calculator. Graph. \(x=4+y^{2}\)

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. $$ f(x)=x^{4}+4 x^{3}-36 x^{2}-160 x+300 $$

Graph. $$ f(x)=\left\\{\begin{array}{ll} -6, & \text { for } x=-3 \\ -x^{2}+5, & \text { for } x \neq-3 \end{array}\right. $$

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. \(f(x)=x^{3}-x\) (Also use algebra to find the zeros of this function.)

Explain why the vertical-line test works.
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