Evaluating Functions

Evaluating functions is a foundational skill in Pre-Algebra that sets students up for success in more advanced mathematical concepts. Whether you're working with linear, quadratic, or polynomial functions, the ability to evaluate a function for a given input is crucial. Let's dive into the steps you need to follow to evaluate functions along with some clear examples to make everything crystal clear.

What is a Function?

Before we jump into evaluating functions, let’s quickly recap what a function is. In mathematical terms, a function is a relation that assigns exactly one output for each input. We often represent functions using function notation, commonly written as \( f(x) \), where \( f \) refers to the function name, and \( x \) represents the input value.

Steps to Evaluate Functions

Evaluating a function involves substituting a specific input into the function and calculating the output. Here are the steps you need to follow:

  1. Identify the Function: Determine the function you’re working with. This could be given in a sentence like “Let \( f(x) = 2x + 3 \).”

  2. Substitute the Input: Take the input value you want to evaluate and substitute it into the function wherever you see \( x \).

  3. Perform the Calculations: Simplify the expression that results from the substitution to find the output.

  4. State the Result: Present your result clearly, stating the output value along with the input value used.

Let’s go through these steps with some examples.

Example 1: Evaluating a Linear Function

Function: \( f(x) = 2x + 3 \)

Input: Let’s evaluate \( f(4) \).

Step 1: Identify the Function.

Here, \( f(x) = 2x + 3 \).

Step 2: Substitute the Input.

We’ll substitute \( 4 \) for \( x \):
\[ f(4) = 2(4) + 3 \]

Step 3: Perform the Calculations.
\[ f(4) = 8 + 3 = 11 \]

Step 4: State the Result.
Thus, when \( x = 4 \), \( f(4) = 11 \).

Example 2: Evaluating a Quadratic Function

Function: \( g(x) = x^2 - 5x + 6 \)

Input: Let’s evaluate \( g(3) \).

Step 1: Identify the Function.

Here, \( g(x) = x^2 - 5x + 6 \).

Step 2: Substitute the Input.

Let’s substitute \( 3 \) for \( x \):
\[ g(3) = (3)^2 - 5(3) + 6 \]

Step 3: Perform the Calculations.
\[ g(3) = 9 - 15 + 6 = 0 \]

Step 4: State the Result.
Therefore, when \( x = 3 \), \( g(3) = 0 \).

Example 3: Evaluating a Piecewise Function

Function:
\[ h(x) = \begin{cases} x + 2 & \text{if } x < 0 \ x^2 & \text{if } x \geq 0 \end{cases} \]

Input: Let’s evaluate \( h(-2) \) and \( h(3) \).

Step 1: Identify the Function.

This function has two cases based on the value of \( x \).

Evaluating \( h(-2) \):
Step 2: Since \(-2 < 0\), we use the first case:
\[ h(-2) = -2 + 2 = 0 \]

Step 3: State the Result.
Thus, when \( x = -2 \), \( h(-2) = 0 \).

Evaluating \( h(3) \):
Step 2: Since \(3 \geq 0\), we use the second case:
\[ h(3) = (3)^2 = 9 \]

Step 3: State the Result.
So, when \( x = 3 \), \( h(3) = 9 \).

Example 4: Evaluating a Function that Involves Absolute Values

Function: \( k(x) = |x - 1| + 5 \)

Input: Let’s evaluate \( k(2) \) and \( k(0) \).

Step 1: Identify the Function.

Here, \( k(x) = |x - 1| + 5 \).

Evaluating \( k(2) \):
Step 2: Substitute \( 2 \):
\[ k(2) = |2 - 1| + 5 \]

Step 3: Perform the Calculations.
\[ k(2) = |1| + 5 = 1 + 5 = 6 \]

Step 4: State the Result.
So, when \( x = 2 \), \( k(2) = 6 \).

Evaluating \( k(0) \):
Step 2: Substitute \( 0 \):
\[ k(0) = |0 - 1| + 5 \]

Step 3: Perform the Calculations.
\[ k(0) = | -1 | + 5 = 1 + 5 = 6 \]

Step 4: State the Result.
Thus, when \( x = 0 \), \( k(0) = 6 \).

Conclusion

Evaluating functions is a straightforward process once you understand the steps involved. Remember to carefully identify the function, substitute the input correctly, perform the calculations accurately, and conclude with the result. Practicing these steps with various types of functions will solidify your understanding and prepare you for more complex mathematical challenges ahead.

Now that you have a solid grasp of how to evaluate functions, you can apply these skills to a variety of mathematical problems in Pre-Algebra and beyond. Happy calculating!