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Chapter 3: Problem 33

Determine whether the equation defines y as a function of \(x .\) \(y=\frac{1}{x}\)

Short Answer

Expert verified
Yes, the equation defines y as a function of x for all x except 0.

Step by step solution

01

Identify the Equation

The given equation is \[ y = \frac{1}{x} n\].
02

Define a Function

Recall that a relation is a function if each input (x-value) corresponds to exactly one output (y-value).
03

Check for Unique Output

For any value of \(x \), the equation \( y = \frac{1}{x} \) uniquely determines the value of \(y \). There is no value of \(x \) that gives more than one value of \(y \).
04

Examine Domain Restrictions

Note that \(x eq 0 \), because division by zero is undefined. However, within its domain (\(x eq 0\)), each \(x \) still corresponds to exactly one \(y \).
05

Conclusion

Since each value of \(x \) (except 0) maps to exactly one value of \(y\), the equation \( y = \frac{1}{x} \) does define \( y \) as a function of \( x \).

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Key Concepts

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

defining a function
In algebra, defining a function is crucial. A function is a specific type of relation where each input, typically denoted as x, corresponds to one and only one output, generally denoted as y. Think of it this way: If you input a value into a machine (our function), it processes it and gives you one specific result. For instance, in the equation \( y = \frac{1}{x} \), for every value of x that you input (excluding x = 0), you will get a unique y-value. Understanding this concept helps you determine if an equation defines a function. It's essential because not every equation forms a function.
domain restrictions
When dealing with functions, you must consider domain restrictions. The domain of a function is the set of all possible input values (x-values) that will provide a valid output. For the function \( y = \frac{1}{x} \), you cannot have x = 0 because dividing by zero is undefined. This specific restriction is crucial in understanding the boundaries of our function.
Always check for any values that make the equation impossible or undefined. If such values exist, exclude them from your domain. In our example:
  • The domain excludes x = 0.
  • Thus, the valid inputs are all real numbers except 0.
unique outputs
A key property of functions is that each input must map to a unique output. In simpler terms, when you put an x into our function, you should get only one corresponding y. For the equation \( y = \frac{1}{x} \), if you choose any x (except 0), you get one and only one result for y. Let's see it step-by-step:
  • If x = 2, \( y = \frac{1}{2} \) (which is 0.5)
  • If x = -3, \( y = \frac{1}{-3} \) (which is approximately -0.333)
You'll notice there's no way to get more than one y-value from a single x-value, making our equation a function.

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