Problem 9 - Calculus with Concepts in Calculus

Question

Find the numerical value of the function at the given values of \(a\). $$ f(x)=\frac{x-1}{x^{2}+4} ; a=2 $$

Step-by-Step Solution

Answer

The numerical value of the function at \( a = 2 \) is \( \frac{1}{8} \).

Step 1: Substitute the given value into the function

Insert the value of \( a = 2 \) into the function \( f(x) \). This means that you will replace every \( x \) in the function with 2. The function becomes: \( f(2) = \frac{2-1}{2^2 + 4} \).

Step 2: Simplify the expression in the numerator

Calculate the expression in the numerator of the fraction. \( 2 - 1 = 1 \). So, the numerator simplifies to 1.

Step 3: Simplify the expression in the denominator

Calculate the expression in the denominator of the fraction. \( 2^2 + 4 = 4 + 4 = 8 \). Thus, the denominator simplifies to 8.

Step 4: Simplify the entire fraction

Now that both the numerator and denominator are simplified, the fraction becomes \( \frac{1}{8} \).

Key Concepts

Numerical Value

The concept of a numerical value is tied to the output you get when you evaluate a function at specific input values. Think of it as the final answer or result of plugging a number into a function. A function like \( f(x) = \frac{x-1}{x^2 + 4} \) can take any number that fits its input requirement (here, any real number).
For instance, to find the numerical value of the function \( f(x) \) when \( a = 2 \), you're tasked with inserting \( 2 \) into the function anywhere there's an \( x \). This process results in a new expression whose numerical value you will calculate. It's like feeding the function with the value to find out what it outputs.

• The numerical value helps in understanding the specific outcome of a function at a particular point.
• It essentially transforms an algebraic expression into a single number, which helps in analyzing real-world quantities easier.

Substituting Values

Substituting values involves replacing the variable in a given expression or function with a specified number. This step is crucial because it allows us to transform a general mathematical statement (which can cover infinite inputs) into a specific one.
To perform substitution, as in the example where \( a = 2 \) is substituted into \( f(x) = \frac{x-1}{x^2 + 4} \), follow these simple steps:

• Identify the variable in the expression (which is \( x \) in this case).
• Replace every occurrence of this variable with the given numerical value (\( 2 \)).
• The function changes from \( f(x) \) to \( f(2) \), resulting in \( \frac{2-1}{2^2 + 4} \).

This substitution helps simplify the expression from a general rule to a specific instance, making numerical evaluation possible. Substitution is widely used not just in functions but also in equations and real-world problem-solving scenarios.

Simplifying Fractions

Simplifying fractions is a key step to arrive at the tidy result of any evaluated expression. It involves reducing a fraction to its simplest form, where the numerator and denominator are as small as possible while still having the same value.
Once substitution has been performed, as seen in \( \frac{2-1}{2^2 + 4} \), you simplify as follows:

• First, resolve operations in the numerator: \( 2 - 1 = 1 \).
• Then, simplify the expression in the denominator: \( 2^2 + 4 = 4 + 4 = 8 \).
• Combine these results to form the fraction \( \frac{1}{8} \).

This step not only ensures clarity but also makes further calculations easier as it's always easier to work with reduced fractions. Factoring, cancelling common terms, and other methods may be used to simplify fractions efficiently in broader contexts.