1. Functions from a Calculus Perspective
Chapter One

2. Functions, Domain and Range
Function: A function is a relation between two variables such that the output can be predicted for every input in the domain.
Domain: The set of possible inputs (usually x-values)
Range: The set of possible outputs (usually y-values)
Different inputs can have the same outputs (y-values can repeat)
One input cannot have multiple outputs (x-values cannot repeat)
The Vertical Line Test is used to determine if a graph represents a function.

3. Identifying Domain and Range from a Graph
Domain: x-values used in drawing the graph
[leftmost value, rightmost value] minus any points of discontinuity
Range: y-values produced when drawing a graph
[lowest value, highest value] minus any points of discontinuity

4. Intercepts y-intercept: x-intercept(s), aka zero(s) of the function,:
point where the graph crosses the y-axis
set x = 0: (0, )
x-intercept(s), aka zero(s) of the function,:
point where the graph crosses the x-axis
set f(x)=0: ( , 0)

5. Even and Odd Functions Even function: Odd function:
symmetric to y-axis
y-axis acts like a mirror (left side is a mirror image of the right side)
f(-x) = f(x)
replace x with –x to get f(-x) then simplify
compare to f(x), if same then even
If (a, b) is on the graph of f(x) then (-a, b) will also be on the graph.
Odd function:
symmetric to origin
origin acts like the axle of a wheel (graph looks the same if you rotate 180°)
f(-x) = -f(x)
multiply f(x) by -1 to get –f(x) then simplify
compare to f(-x), if same then odd
If (a, b) is on the graph of f(x) then (-a, -b) will also be on the graph.

6. Continuity
Continuous function: The graph can be drawn from -∞ to +∞ without lifting the pen/pencil and there are no sharp turns.
Examples of continuous functions:
(constant, linear, quadratic, other polynomials, some rational)
Examples of non-continuous functions:
(absolute value, piecewise, many rational)
Types of discontinuity
Removable discontinuity: hole in the graph (division by zero)
Infinite discontinuity: output values shoot up to positive infinity or down to negative infinity (also division be zero)
Jump discontinuity: There is a jump up or down from one part of the graph to another (piecewise functions)

7. Graphing piecewise functions on a TI-83+ calculator
y = (Rule 1)(Condition 1) + (Rule 2)(Condition 2) + …
Inequality symbols are found in the TEST menu which can be reached by pressing 2nd MATH
May be useful to switch calculator from connected to dot mode.

8. Intermediate Value Theorem
If f(x) is a continuous function and a < b then f(x) is equal to every y-value between f(a) and f(b) at least once.
This theorem is most useful when the signs of f(a) and f(b) are opposites. In this case, we know that the graph of the function crossed the x-axis at least once in the interval (a, b).

9. End Behavior
What’s happening to f(x) values on the left end of the graph (i.e.: as x-values approach negative infinity)?
are they going up (approaching positive infinity)?
are they going down (approaching negative infinity)?
are they leveling off (approaching a certain value)?
What’s happening to f(x) values on the right end of the graph (i.e.: as x-values approach positive infinity)?

10. Increasing/Decreasing/Constant Intervals
Increasing Interval: y-values are going up from Left to Right
Decreasing Interval: y-values are going down from L. to R.
Constant Interval: y-values remain the same from L. to R.
x-values are always used to describe intervals

11. Relative Extrema Relative maximum: highest point in an interval.
Relative minimum: lowest point in an interval.
points are used to describe max and mins

12. Average Rate of Change
Use the slope formula and the beginning and ending x-values for a given interval.
Slope formula

13. Parent Graphs f(x) = c constant f(x) = x linear (identity)
f(x) = x2 quadratic
f(x) = x3 cubic
f(x) = | x | absolute value
f(x) = square root
greatest integer
f(x) = 1/x rational (reciprocal)

14. Transformations of a parent graph g(x) = a f(bx + c) + d
Sequence of Transformations:
c, b, a, d a
Vertical expansion if |a| > 1
Vertical compression if |a| < 1
Vertical reflection if a < 0 b
Horizontal compression if |b| > 1
Horizontal expansion if |b| < 1
Horizontal reflection if b < 0 c
Horizontal shift: left if c > 0, right if c < 0 d
Vertical shift: up if d > 0, down if d < 0
Generalizations:
Inside affect horizontal opposite of what you expect
Outside affects vertical the way you expect
Multiply affects expansion/compression
Mult by neg affects reflection
Add/Subtract affects shift

15. Function Operations (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x)
(f ÷ g)(x) = f(x) ÷ g(x); g(x) ≠ 0
Recall that f(x) and g(x) are the outputs or y-values for a given x-value.
Therefore (f + g)(x) means to add the two y-values associated with the given x-value.
Function Operations: Combine two outputs.

16. Composition of Functions
(f º g)(x) = f( g( x ) )
Input x into g(x)
find the output for g(x)
input the result into f(x)
find the output for f(x)
The output of the inner function becomes the input of the outer function.

17. Inverse Functions The inputs and the outputs switch roles.
the input of f(x) becomes the output of f-1(x)
the output of f(x) becomes the input of f-1(x)
If (a,b) is on the graph of f(x), then (b,a) will be on the graph of f-1(x)
Horizontal Line Test
If it works, then the function is a one-to-one function and its inverse will be a function.
If it does not work, then the function is not a one-to-one function and the inverse will not be a function.

18. Verifying Inverse Functions
f and g are inverses if and only if f(g(x)) = g(f(x)) = x
Find the composite of f(g(x)).
If not equal to x, then f and g are not inverses
If equal to x, then continue by finding composite of g(f(x))
If equal to x, then f and g are inverses.

19. Finding Inverses Algebraically
Replace f(x) with y.
Switch x and y.
Solve for y.
Replace y with f-1(x).