Math Lesson 16.1.8 - Evaluating Functions

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Welcome to our Math lesson on Evaluating Functions, this is the eighth lesson of our suite of math lessons covering the topic of Relation and Function, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Evaluating Functions

Evaluating a function means finding the images (y-values) of certain values of the independent variable (i.e. of x-values) by substituting the values of the independent variable x in the formula of the function. In this way, we find the corresponding y-values, which are the images required. Let's explain this point through an example.

Example 7

Find the image of -1, 2 and 3 of the functions shown below.

1. x → -x3 + 1/x
2. x → (2 + x)/x

Solution 7

The image of a value in a function is found by substituting that value in the formula of that function. Thus, we have

1. x → -x³ + 1/x
   f(x) = -x³ + 1/x
   The image of -1 [i.e. f(x) for x = -1] is
   f(-1) = -(-1)³ + 1/-1
   = -(-1) + 1/-1
   = 1 - 1
   = 0
   The image of 2 [i.e. f(x) for x = 2] is
   f(2) = -2³ + 1/2
   = -8 + 1/2
   = -16/2 + 1/2
   = -15/2
   The image of 3 [i.e. f(x) for x = 3] is
   f(2) = -3³ + 1/3
   = -27 + 1/3
   = -81/3 + 1/3
   = -80/3

It is worth highlighting the fact that the ordered pairs determined by the formula of a function represent only a part of the Cartesian product X × Y. This Cartesian product represents all possible ordered pairs where the first number in each pair is from the set X and the second number from the set Y. However, not all these pairs are values (give true results) of a given function. For example, the ordered pair (3, 1) belongs to the function f(x) = 2x - 5 because f(x) = 2 · 3 - 5 = 6 - 5 = 1, which is the same value as the second number of the given ordered pair, but (1, 4) does not belong to this function as f(x) = 2 · 1 - 5 = 2 - 5 = -3. This result is different from the second value of the given ordered pair (4).

You have reached the end of Math lesson 16.1.8 Evaluating Functions. There are 9 lessons in this physics tutorial covering Relation and Function, you can access all the lessons from this tutorial below.

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6. Continuing learning functions - read our next math tutorial: Injective, Surjective and Bijective Functions. Graphs of Functions

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