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How do you evaluate expressions given functions?

Published in Function Evaluation 2 mins read

Evaluating expressions given functions involves substituting a specific value or expression for the variable in the function and then simplifying the resulting expression.

Understanding Function Evaluation

The core idea is to replace the function's variable (often x) with the given value or expression. This process allows you to determine the function's output for that specific input.

Steps to Evaluate Expressions Given Functions

  1. Identify the Function: Clearly understand the function's definition, such as f(x) = 2x + 4.
  2. Identify the Input: Determine the value or expression that needs to be substituted for the variable (e.g., evaluate f(5)).
  3. Substitute: Replace the variable in the function's expression with the given input. For example, in f(x) = 2x + 4, to find f(5), you would replace x with 5, resulting in f(5) = 2(5) + 4.
  4. Simplify: Perform any necessary arithmetic operations to simplify the expression. Continuing the example, f(5) = 2(5) + 4 = 10 + 4 = 14.

Example: Evaluating f(x) = 2x + 4 at x = 5

As mentioned in the reference material, let's evaluate the function f(x) = 2x + 4 when x = 5.

  • Function: f(x) = 2x + 4
  • Input: x = 5
  • Substitution: f(5) = 2(5) + 4
  • Simplification: f(5) = 10 + 4 = 14

Therefore, f(5) = 14.

Evaluating with Expressions

Function evaluation isn't limited to numerical values. You can also substitute expressions.

Example:

Let g(x) = x2 + 1. Evaluate g(a + 1).

  1. Function: g(x) = x2 + 1
  2. Input: x = a + 1
  3. Substitution: g(a + 1) = (a + 1)2 + 1
  4. Simplification: g(a + 1) = a2 + 2a + 1 + 1 = a2 + 2a + 2

Evaluating Combined Functions

Sometimes, you might encounter problems involving combinations of functions. You would evaluate each function individually and then combine the results as indicated by the problem.

Example:

If f(x) = x + 2 and g(x) = 3x, find f(2) + g(2).

  1. Evaluate f(2): f(2) = 2 + 2 = 4
  2. Evaluate g(2): g(2) = 3(2) = 6
  3. Combine: f(2) + g(2) = 4 + 6 = 10

Summary Table

Step Description Example (f(x) = x2, evaluate f(3))
1. Identify Recognize the function and the value to substitute. f(x) = x2, evaluate f(3)
2. Substitute Replace the function's variable with the given value or expression. f(3) = (3)2
3. Simplify Perform any arithmetic operations to find the function's output for that input. f(3) = 9

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