1. Warm-up (8 min.) Find the domain and range of of f(x) = .
State whether the function is odd, even, or neither. Confirm algebraically.
f(x) = 2x4 b. g(x) = 2x3 – 3x c. h(x) =
Find all horizontal and vertical asymptotes.

2. 1.3 Twelve Basic Functions 1.5 Transformations of Functions
After today’s lesson you will be able to:
Recognize graphs of twelve basic functions
Determine domains of functions related to the twelve basic functions
Combine the twelve basic functions in various ways to create new functions
To transform the basic functions to describe other functions

3. The Linear (Identity) Function
f(x) = x
Domain and Range: All real numbers (-,)
Always continuous and Always increasing (-,)
Unbounded
Symmetry through the origin (odd function)
No asymptotes nor extrema
End behavior:
How can you the identity function to describe g(x) = -x?

4. The Quadratic (Squaring) Function
f(x) = x2 Note: Graph called parabola.
Domain: All real numbers (-,)
Range: All non-negative real numbers [0,)
Always continuous (-,)
Decreasing: (-, 0) Increasing: (0,)
Bounded below at (0,0).
Symmetry through the y-axis (even function)
No asymptotes
End behavior:
Absolute minimum at (0,0)
How can you use the squaring function to describe g(x) = x2 – 2?

5. The Cubic (Cubing) Function
f(x) = x3 Note: The graph changes curvature at the origin which is called a point of inflection.
Domain and Range: All real numbers (-,)
Always continuous (-,)
Always Increasing: (-,)
Unbounded
Symmetry through the origin (odd function)
No asymptotes and no extrema
End behavior:
Describe the graph of h(x) = (x+3)3 using f(x)?

6. The Square Root Function
f(x) = x Graph is top-half of a sideways parabola.
Domain & range: All non-negative real numbers [0,)
Always continuous and always increasing: [0,)
Bounded below at (0,0).
No symmetry and No asymptotes
End behavior:
Absolute minimum at (0,0)
Use the square root function to describe g(x) = x-2

7. The Common Logarithmic Function
f(x) = log x
Domain: All positive real numbers (0,)
Range: All real numbers (-,)
Always continuous and always increasing: (-,)
Bounded on the left by x = 0 (y-axis)
Note: Inverse of exponential function f(x) =10x.
No symmetry and no extrema
End behavior:
How could you use the log x function to describe g(x) = log (x – 2) + 3?

8. The Natural Logarithmic Function
f(x) = ln x Used to describe many real-life phenomena included intensity of earthquakes (Richter scale).
Domain: All positive real numbers (0,)
Range: All real numbers (-,)
Always continuous and always increasing: (-,)
Bounded on the left by x = 0 (y-axis)
Note: Inverse of exponential function f(x) =ex.
No symmetry and no extrema
End behavior:
Where are the vertical and horizontal asymptotes of
g(x) = ln (x + 1) -2?

9. The Exponential Function
f(x) = ex Graph is half of a hyperbola.
Value of e 
Domain: All real numbers (-,)
Range: All positive real numbers (0,)
Always continuous and always increasing: (-,)
Bounded below by y = 0 (x-axis)
Note: Inverse of exponential function f(x) = ln x.
No symmetry and no extrema
End behavior:
Use the exponential function to describe g(x) = -ex.

10. The Absolute Value Function
f(x) = |x| Note: V-shaped graph
Domain: All real numbers (-,)
Range: All non-negative real numbers [0,)
Always continuous (-,)
Decreasing: (-, 0) Increasing: (0,)
Bounded below at (0,0).
Symmetry through the y-axis (even function)
No asymptotes
End behavior:
Absolute minimum at (0,0)
Use the graph of the absolute value function to describe the graph of g(x) = |x | + 2 .

11. The Reciprocal Function
f(x) = 1/x Note: Graph called a hyperbola
Domain & Range: All real numbers  0. (-,0)(0,)
Infinite discontinuity at x = 0
Unbounded
Symmetry through the origin (odd function)
V.A.: x = 0 (y-axis) H.A. y = 0 (x-axis)
End behavior:
No extrema
Write an equation for g(x) which transforms the reciprocal function 2 units to the left?

12. The Greatest Integer (Step) function
f(x) = [x] = int (x) Also called the “step” function.
Domain: All real numbers (-,)
Range: All integers
Discontinuous at each integer (jump discontinuity)
Constant between integer values
Unbounded and no symmetry
No asymptotes and no extrema
End behavior:

13. The Sine Function f(x) = sin x Note: Graph is a continuous wave
Domain: All real numbers (-,)
Range: [-1,1]
Always continuous (-,)
Increasing: (0, p/2)(3p/2, 2p) Decreasing: (p/2,3p/2)
Note: Periodic function (repeats every 2p units).
Bounded
No asymptotes but symmetry through the origin (odd function)
Local Maximum: p/2 + kp, where k is an odd integer
Local Minimum: 3p/2 + kp, where k is an integer
Describe the change in the graph of f(x) = sin x to create the graph of g(x) = 2 sin x.

14. The Cosine Function f(x) = cos x Note: Graph is a continuous wave
Domain: All real numbers (-,)
Range: [-1,1]
Always continuous (-,)
Decreasing: (0, p) Increasing: (p,2p)
Note: Periodic function (repeats every 2p units).
Bounded
No asymptotes but symmetry through the y-axis (even function)
Local Maximum: kp, where k is an even integer
Local Minimum: kp/2, where k is an odd integer
Write the equation of a cosine function with maximum of 5 and minimum of -5 with no horizontal shift.

15. The Logistic Function
f(x) = Note: Model for many applications of biology and business.
Domain: All real numbers (-,) Range: (0,1)
Always continuous and always increasing (-,)
Bounded (below with x-axis (y = 0 and above by y = 1)
Symmetry through the point (0,1/2)
No extrema
H.A.: y = 0 and y = 1
End behavior

16. Infinite Discontinuity
Ex1 Which of the basic functions are continuous? For the functions that are discontinuous, identify as infinite or jump.
Continuous
Jump Discontinuity
Infinite Discontinuity

17. Ex 2 Which of the basic functions have symmetry? Describe each.
Symmetry in
x-axis
y-axis
(even)
Symmetry in origin
(odd)

18. Exploration: Looking for Asymptotes
Two of the basic functions have vertical asymptotes at x = 0. Which two?
Form a new function by adding these functions together. Does the new function have a vertical asymptote at x = 0?
Three of the basic functions have horizontal asymptotes at y = 0. which three?
Form a new function by adding these functions together. Does the new function have a horizontal asymptote y = 0?
Graph f(x) = 1/x, g(x) = 1/(2x2-x), and h(x) = f(x) + g(x). Does h(x) have a vertical asymptote at x = 0? Explain.

19. Piecewise-defined Functions
Which of the twelve basic functions has the following piecewise definition over separate intervals of its domain?
x if x  0
f(x) =
-x if x < 0

20. Ex 3 Use the basic functions from this lesson to construct a piecewise definition for the function shown. Is your function continuous?

21. Ex4 To what basic functions does w(x) = x3 – 2x2 + x relate?
Describe the behavior of the function above.

22. Tonight’s Assignment
Complete 1.3 & 1.5 Assignments from Unit 1 outline
Unit 1 Test will be Friday, Feb. 5th