Problem 20 - Brief Applied Calculus

Question

For each function: $$ f(x)=\frac{1}{\sqrt{x}} ; \text { find } f(4) $$

Step-by-Step Solution

Answer

The value of \( f(4) \) is \( \frac{1}{2} \).

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{\sqrt{x}} \). This means that for any input \( x \), you need to take the square root of \( x \), and then find the reciprocal of that square root.

Step 2: Substitute the Given Value

We need to find \( f(4) \). To do this, replace \( x \) in the function equation with 4. So, \( f(4) = \frac{1}{\sqrt{4}} \).

Step 3: Calculate the Square Root

Find the square root of 4. The square root of 4 is 2, so \( \sqrt{4} = 2 \).

Step 4: Find the Reciprocal

Now, find the reciprocal of 2. The reciprocal of 2 is \( \frac{1}{2} \).

Step 5: Arrive at the Final Answer

Thus, \( f(4) = \frac{1}{2} \). This means when \( x = 4 \), the output of the function is \( \frac{1}{2} \).

Key Concepts

Understanding Reciprocals

A reciprocal is simply what you need to multiply a number by to get 1. For example, if you're dealing with the number 5, its reciprocal is \( \frac{1}{5} \), because \( 5 \times \frac{1}{5} = 1 \). This concept is particularly useful in various areas of mathematics, especially when working with fractions or ratios.

In function evaluation, finding the reciprocal helps us determine how certain operations, like division, can be undone. For instance, finding the reciprocal of the square root, as seen in the example \( f(x) = \frac{1}{\sqrt{x}} \), means that you're reversing the effect of the square root by turning it into a fraction over 1. This ability to "flip" numbers is essential when simplifying mathematical expressions or solving equations.

• Reciprocal of a number \( a \) is \( \frac{1}{a} \).
• Helps in simplifying fractions and solving equations.
• In function evaluation, it reverses the effect of multiplication or division with respect to 1.

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). It's denoted by the radical symbol \( \sqrt{} \).

In our original function \( f(x) = \frac{1}{\sqrt{x}} \), calculating the square root is a critical step. By understanding square roots, we determine half of the operations needed to evaluate the function at a specific point. Knowing square roots makes problems related to quadratic equations, area calculations, and geometric applications much easier.

• Square root of \( x \) is denoted as \( \sqrt{x} \).
• Square root \( \sqrt{x} \) gives us a number which, when squared, equals \( x \).
• Crucial for understanding and working with functions involving radicals.

Applying the Substitution Method

The substitution method is a fundamental technique used to evaluate functions at specific points. It involves replacing variables within an expression or function with specific values to simplify and solve it. This approach is widely used not only in basic arithmetic and algebra but also in calculus and more advanced math topics.

In the problem we were given, substitution was performed by replacing \( x \) with 4 in our function \( \frac{1}{\sqrt{x}} \). This step is crucial because it allows us to determine the output for a specific input, making complex functions manageable and understandable by breaking them into simpler parts.

• Replace the variable with the given specific value.
• Simplify the expression to obtain results.
• Widely applicable across various mathematical concepts.