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Chapter 3: Problem 93

Find the domain of \(f\). $$f(x)=\frac{2 x-7}{5}$$

Short Answer

Expert verified
The domain of the function is all real numbers.

Step by step solution

01

- Understand the function

Examine the given function: \( f(x) = \frac{2x-7}{5} \). This is a rational function.
02

- Identify restrictions for the domain

Since the function is a rational expression, we need to determine any restrictions. For rational functions, the denominator must not equal zero. However, in this case, the denominator (5) is a constant and never zero.
03

- Determine the domain

Since there are no values that will make the denominator zero or any other restrictions, the function is defined for all real numbers. Therefore, the domain of the function is all real numbers.

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Key Concepts

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

Rational Functions
A rational function is a type of function represented by the quotient of two polynomials. In simpler terms, it's a fraction where the numerator and the denominator are both polynomials. For example, in the function given in the exercise, \( f(x) = \frac{2x-7}{5} \), both the numerator \(2x-7\) and the denominator \(5\) are polynomials. Because there's a fraction involved, rational functions can exhibit interesting behaviors, like asymptotes and holes, depending on the values of the variables involved. Understanding these kinds of functions is crucial, especially when considering their domains and any potential restrictions.
Domain Restrictions
When dealing with rational functions, one of the key considerations is identifying the domain, meaning the set of all possible input values (\(x\)-values) that the function can accept. For a rational function, the main restriction involves the denominator, as the denominator must never be zero. If the denominator were zero, the function would be undefined because you cannot divide by zero. In our example function, \( f(x) = \frac{2x-7}{5} \), the denominator is a constant value of 5, which is never zero. This means there are no restrictions on the domain, and the function can accept any real number as an input.
Real Numbers
Real numbers include all the numbers that can be found on the number line. This includes both rational numbers (such as 3, \(-\frac{1}{2}\), and 0.75) and irrational numbers (such as \(\frac{\text{pi}}{2}\), and \(\text{sqrt}2\)). When we say that a function's domain includes all real numbers, it means that any number from this set can be plugged into the function. For the function \( f(x) = \frac{2x-7}{5} \), since its denominator is always a non-zero constant, you can use any real number as an input. So, the domain is all real numbers.

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Most popular questions from this chapter

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