Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\sqrt{x^{2}-4}\) \(x=-2\)

Step-by-Step Solution

Answer

The domain of the function is \(-\infty \leq x \leq -2\) and \(2 \leq x \leq \infty\), the range is \(0 \leq f(x) < \infty\), and \(f(-2)=0\).

Step 1: Find the Domain

The inside of a square root must always be greater than or equal to zero, because we cannot take the square root of a negative number. Therefore, we have \(x^{2}-4 \geq 0\). To solve this inequality, express it as \(x^{2}\geq 4\). This yields two solutions, \(x \leq -2\) or \(x \geq 2\). Thus, the domain of the function is \(-\infty \leq x \leq -2\) and \(2 \leq x \leq \infty\).

Step 2: Find the Range

Square root functions yield output values greater than or equal to zero. In this case, as the minimal value the function \(x^{2}-4\) can reach is 0, the minimal number the function \(\sqrt{x^{2}-4}\) can reach is \(\sqrt{0} = 0\). Hence, the range is \(0 \leq f(x) < \infty\).

Step 3: Evaluate \(f(x)\) at \(x=-2\)

We can substitute \(x=-2\) in \(f(x)=\sqrt{x^{2}-4}\r). The function becomes \(f(-2)=\sqrt{(-2)^{2}-4}=\sqrt{4-4}=0\). Therefore, \(f(-2)=0\).

Key Concepts

Square Root Functions

Square root functions involve taking the square root of an expression. These functions have a unique property: the inside of the square root, called the radicand, must be non-negative.
This is to ensure that the square root is a real number, as negative square roots are not defined in the set of real numbers. For our function, \(f(x) = \sqrt{x^2 - 4}\), the radicand is \(x^2 - 4\).
We find that the inside of the square root must be greater than or equal to zero, leading to inequalities that help us find the domain. In this case, to keep \(x^2 - 4\) non-negative, the domain is either \(x \leq -2\) or \(x \geq 2\).
Square root functions also tend to have certain ranges, starting from zero to positive infinity, matching the outputs of the square roots themselves.
Understanding square root functions is key to determining their domain and range, which is essential for plotting these functions accurately.

Inequalities

Inequalities are statements about the relative size of two values. They are essential in determining where functions are defined.
When working with square roots, like in \(f(x) = \sqrt{x^2 -4}\), you must ensure that the expression under the square root is non-negative.

• Start with the inequality \(x^2 - 4 \geq 0\).
• Resolve it by finding points where the expression equals zero, such as \(x^2 = 4\).
• Solve for \(x\), giving \(x = -2\) or \(x = 2\).

When an inequality involves a square, it often splits into two parts.
For \(x^2 - 4\), it delivers the solutions \(x \leq -2\) and \(x \geq 2\). These solutions aid us in figuring out the domain by specifying where the function "lives" on a number line. Understanding inequalities ensures that functions remain valid and respect the mathematical logic needed to find their solutions.

Function Evaluation

Function evaluation involves substituting a specific value into a function to get an output. It's like asking, "What does my function say when \(x\) equals this?"
In our case, we need to evaluate \(f(x) = \sqrt{x^2 -4}\) at \(x = -2\).
To do this, plug \(-2\) into the function: \[f(-2) = \sqrt{(-2)^2 - 4} = \sqrt{4 - 4} = \sqrt{0} = 0\]. This process of substituting helps determine particular points on the graph of the function.
It also checks whether given values fall within the domain of the function. Here, since \(f(-2) = 0\), it illustrates how the domain supports this input, confirming correctness.
Evaluating functions is crucial for understanding specific outputs, validating them against the domain, and interpreting the graphical behavior of the function.