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Chapter 3: Problem 56

Find the domain of each function. \(h(x)=\frac{2 x}{x^{2}-4}\)

Short Answer

Expert verified
The domain is all real numbers except \( x = 2 \) and \( x = -2 \).

Step by step solution

01

Identify the restricted values in the denominator

The domain of a function is restricted where the denominator is zero. Therefore, solve the equation in the denominator for zero: \( x^2 - 4 = 0 \)
02

Solve for the restricted x-values

Factor the denominator expression to find the values of \( x \) that make the denominator zero.\( x^2 - 4 = (x - 2)(x + 2) = 0 \)Hence, the solutions are \( x = 2 \) and \( x = -2 \).
03

Define the domain

The domain of the function \( h(x) = \frac{2x}{x^2 - 4} \) includes all real numbers except the values that make the denominator zero. Therefore, the domain of the function is all real numbers except \( x = 2 \) and \( x = -2 \).

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Key Concepts

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

Restricted Values
To find the domain of a function, we first need to identify any restricted values.
Restricted values are specific values that the variable, often x, can't take because they make the function undefined or invalid.
In the context of functions with denominators, we look for values that would make the denominator equal to zero, as division by zero is undefined.
Therefore, our first step in solving this problem is determining where the denominator is zero.
Denominator Zero
The denominator of a fraction cannot be zero because it would make the fraction undefined.
To find these restricted values, we set the denominator equal to zero and solve for x.
For this exercise, the denominator is given as \(x^2 - 4\).
This means we solve the equation: \[ x^2 - 4 = 0 \] This equation tells us the values of x that make the denominator zero.
Next, we'll factor the quadratic expression.
Factoring
Factoring is a method used to break down a complex expression into simpler components that are multiplied together.
For the given quadratic expression \(x^2 - 4\), we notice it can be factored into two binomials: \[ x^2 - 4 = (x - 2)(x + 2) \]
Now we set each factor equal to zero to find the restricted values:
(x - 2) = 0 x = 2 (x + 2) = 0 x = -2 These are our restricted values because they make the denominator of the function zero.
Therefore, in our function \(h(x) = \frac{2x}{x^2 - 4}\) the domain excludes \(x = 2\) and \(x = -2\). The final domain can be written as all real numbers except \(x = 2\) and \(x = -2\).

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