y = f(x)

1) What does ƒ(x) mean?

Whatever is inside of the parentheses is the input (or x) value, and ƒ(x) is the output (or y) value.

If ƒ(x) = x³, then ƒ(2) = 8 since 2³ = 8

For any input (a, 16, or x²), plug it in each time for x.

If ƒ(x) = (3x)³ - x:

1. ƒ(a) = (3a)³ - a
2. ƒ(16 ) = (3 * 16) ³ - 16
3. ƒ(x²) = (3(x²))³ - x²

2) Find values on a graph of y = ƒ(x)

In order to determine ƒ(x) values from a graph, start on the x-axis, and trace up or down from the input x value until you hit the graph. Then move perpendicularly to the y-axis to find the y value of ƒ(x).

2-1. Practice:

Use the graph below to determine the value of ƒ(1).

1. 0
2. 1
3. 2
4. 3
5. 4

3) Function Shifts

• ƒ(x + c) shifts the function in the opposite direction of c.
  1. ƒ(x + 5) creates a shift to the LEFT (opposite of the +5)
  2. ƒ(x - 5) creates a shift to the RIGHT (opposite of the -5)
• ƒ(x) + c shifts the function in the same direction of c.
  1. ƒ(x) + 5 creates a UPWARD shift (same as the +5)
  2. ƒ(x) - 5 creates a DOWNWARD shift (same as the -5)

4) Domain and Range

The domain of a function is all of the possible inputs that the function can accept.
For example, if y = 1 ⁄ (x - 3)², the domain would equal all values except 3.

The range of a function is all the possible outputs the function can produce.
In the example above the range would also be limited to all values greater than or equal to zero.

Determine the Domain and Range for the following three functions:

1. y = |x|
2. y = x ⁄ (x + 6)
3. y = - (-x)^(1⁄₂)