Problem 20 - Applied Calculus

Question

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. $$ f(x)=\frac{1}{\sqrt{x}} ; \text { find } f(4) $$

Step-by-Step Solution

Answer

f(4) = \frac{1}{2}; Domain: (0, \infty); Range: (0, \infty).

Step 1: Evaluate the function at the given point

To find \( f(4) \), we need to substitute \( x = 4 \) into the function \( f(x) = \frac{1}{\sqrt{x}} \). So, \( f(4) = \frac{1}{\sqrt{4}} = \frac{1}{2} \).

Step 2: Determine the domain of the function

The function \( f(x) = \frac{1}{\sqrt{x}} \) involves a square root in the denominator, which means \( x \) must be positive to have a valid square root (\( x > 0 \)). Therefore, the domain of \( f(x) \) is all positive real numbers, expressed as \( (0, \infty) \).

Step 3: Determine the range of the function

Given \( f(x) = \frac{1}{\sqrt{x}} \), as \( x \) approaches 0 from the positive side, \( \sqrt{x} \) becomes very small, thus \( f(x) \) becomes very large. As \( x \) increases, \( \sqrt{x} \) increases, making \( f(x) \) smaller but always positive. Therefore, the range is all positive real numbers \((0, \infty)\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a given input. For instance, in our exercise, we wanted to find the value of \( f(x) = \frac{1}{\sqrt{x}} \) when \( x = 4 \). This is done by simply replacing \( x \) with \( 4 \) in the expression:

• Substitute the number: \( f(4) = \frac{1}{\sqrt{4}} \).
• Simplify the expression to get the result: \( f(4) = \frac{1}{2} \).

This evaluation shows that the function \( f(x) \) generates an output of \( \frac{1}{2} \) when \( x \) equals 4. The key is to replace \( x \) in the function with the specific value you want to evaluate and simplify accordingly. Understanding function evaluation is essential as it allows you to determine the output of functions for various inputs.
Next time, be sure to carefully substitute and compute the expression step-by-step for accuracy.

Domain of a Function

The domain of a function is all the possible input values \( x \) that can be used in the function without making it undefined. For the function \( f(x) = \frac{1}{\sqrt{x}} \), there is a square root in the denominator.
This means we can't have a zero or negative number in the denominator because:

• Square roots of negative numbers are not real numbers.
• A zero in the denominator makes the function undefined.

Thus, \( x \) must be greater than 0 to ensure the square root is valid and not zero. Therefore, the domain is all positive real numbers, written as \((0, \infty)\).
Whenever you determine the domain, always consider the restrictions placed by the operations in the function like square roots, fractions, etc.

Range of a Function

The range of a function consists of all the possible output values. For \( f(x) = \frac{1}{\sqrt{x}} \), we need to consider how the output behaves based on the possible input values from the domain.
Since inputs \( x \) are positive numbers (as we determined the domain to be \((0, \infty)\)), the outputs are going to be:

• Large when \( x \) is close to 0, because \( \frac{1}{\sqrt{x}} \) increases as \( \sqrt{x} \) gets smaller.
• Small and closer to zero when \( x \) is very large, as \( \sqrt{x} \) becomes larger, reducing the fraction’s value.

The outputs are always positive since the square root of a positive number is positive. Hence, our range for \( f(x) \) is all positive real numbers, also expressed as \((0, \infty)\).
To find the range, analyze how changes in the domain affect the output values. This involves looking at how an increase or decrease in \( x \) alters the behavior of the function.