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Mathematics LibreTexts

1.3: Function Notation and Simplify Expressions

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The goal of this section is to practice function notation. Functions have many interconnected mathematical concepts. Therefore, more detail about functions will emerge in your Precalculus course. This section will practice the correspondence of inputs to outputs using function notation, giving further practice with the properties introduced in previous sections.

The following three equations express three different mathematical rules. The rule creates a correspondence between x-values and y-values.

Equation The Rule In Words
y=2x+6 Double the x-value then add 6.
y=x24 Take half of the x-value then subtract 4.
y=x210 Square the x-value then subtract 10.
Definition: Function

A function is a rule that assigns each input to exactly one output

A table displays the correspondence between inputs (x-values) and outputs (y-values). The three functions above are shown below using x-y tables. An x-value is chosen, the rule is followed, and a y-value results. Each y-value is computed according to the function’s rule.

x y=2x+6
10 2(10)+6=14
3 2(3)+6=0
0 2(0)+6=6
2 2(2)+6=10
x y=x24
6 6÷24=7
0 0÷24=4
6 6÷24=1
18 18÷24=5
x y=x210
1 (1)210=9
0 0210=10
1 1210=9
4 4210=6

Since the output of the function depends on the input, the y-value is called the dependent variable. Function notation emphasizes this dependence by replacing the variable y with f(x).

x-y notation Function notation Read out loud
y=2x+6 f(x)=2x+6 f of x equals two x plus six”
Example 1.3.1

Let f(x)=2x+6. Evaluate of each of the following.

  1. f(1)
  2. f(12)
  3. f(3a)

Solution

In each problem, the input, or x-value, is given inside the parentheses. We substitute the given x-value into the function f(x)=2x+6.

  1. f(1)=2(1)+6=2+6=4. Therefore, f(1)=4.
  2. f(12)=2(12)+6=1+6=7. Therefore, f(12)=7.
  3. The x-value is replaced by 3a. This will yield an expression rather than a numerical value: f(3a)=2(3a)+6=6a+6. Therefore, f(3a)=6a+6.
Definition: Naming Functions f, g, and h

Functions are typically named with the letters f, g, and h, but functions can take on other letter-names as well. Using different letter-names for functions can help us refer to several different functions in one problem. Example 1.3.2 makes use of this concept.

Example 1.3.2

Let h(x)=f(x)+g(x) where f(x)=3x2 and g(x)=5x9. Evaluate each of the following:

  1. h(4)
  2. h(12)
  3. h(a+1)

Solution

The function h is defined as the sum of f and g. Plug the given input-value into each function, evaluate to find each output, and take the sum.

  1. h(4)=f(4)+g(4)=3(4)2f(4)+5(4)9g(4)=316+(209)=4829=19

Therefore, h(4)=19.

  1. h(12)=f(12)+g(12)=3(12)2f(12)+5(12)9g(12)=3(14)+529=34+104364=234

Therefore, h(12)=234

  1. Replace x with a+1, then simplify. h(a+1)=f(a+1)+g(a+1)

=3(a+1)2f(a+1)+5(a+1)9g(a+1)=3(a+1)(a+1)+5a+59=3(a2+2a+1)+5a4=3a2+6a+3+5a4=3a2+6a+5a+34h(a+1)=3a2+11a1

Example 1.3.3

Let f(x)=2x24x+1. Evaluate each of the following:

  1. f(3)
  2. f(a+2)
  3. f(a+2)f(a)

Solution

  1. Replace each x with x=3.

f(3)=2(3)24(3)+1=29+12+1=18+12+1=31

Therefore, f(3)=31.

  1. Replace each x with a+2. Then simplify.

f(a+2)=2(a+2)24(a+2)+1=2(a2+4a+4)4a8+1Use Special Products and the Distributive Property=2a2+8a+84a8+1=2a2+8a4a+88+1Rearrange terms to combine like termsf(a+2)=2a2+4a+1

  1. We will subtract f(a) from the answer to part b.

2a2+4a+1f(a+2)(2a24a+1)f(a)Use answer from part b: f(a+2)=2a2+4a+1=2a2+4a+12a2+4a1Subtract a quantity, so subtract each term.=2a22a2+4a+4a+11Rearrange terms to combine like terms.

Answer f(a+2)f(a)=8a

Example 1.3.4

Let f(x)=3x4. Evaluate each of the following.

  1. f(a2)
  2. [f(a)]2

Solution

The order of operations will yield different results.

  1. f(a2) indicates x=a2. Therefore, f(a2)=3a24.
  2. [f(a)]2 indicates x=a. After substitution, square the result.

[f(a)]2=(3a4)2Substitute x=a into the function f.[f(a)]2=9a224a+16Use the Special Products Formula.

Try It! (Exercises)

For #1-13, evaluate the functions using f and g as defined below. Do not use a calculator.

f(x)=2x+1

g(x)=x2+3x2

  1. f(3)
  2. g(1)
  3. f(12)
  4. g(2)
  5. f(1)g(3)
  6. g(2a)
  7. f(3a)
  8. f(a4)
  9. g(a1)
  10. g(a2)
  11. [f(a)]2
  12. g(a)f(a+1)
  13. g(a)f(a2)

14. Let h(x)=f(x)+g(x). Evaluate each of the following:

  1. h(5)
  2. h(4a)
  3. h(a3)

For #15-34, evaluate the functions using f and g as defined below. Do not use a calculator.

f(x)=xx+1

g(x)=4xx+2

  1. f(3)
  2. f(3)
  3. g(2)
  4. g(2)
  5. f(2)+g(4)
  6. f(0)g(6)
  7. [f(1)]2
  8. [g(3)]2
  9. f(12)
  10. g(12)
  11. f(a1)
  12. g(a1)
  13. [g(1)]2
  14. [f(4)]2
  15. [f(2)]1
  16. [g(1)]1
  17. [g(14)]1
  18. [f(5)]1+[g(5)]1
  19. [f(5)+g(5)]1
  20. f(a)g(a)

This page titled 1.3: Function Notation and Simplify Expressions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.

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