1. Library of Functions
You should be familiar with the shapes of these basic functions. We'll learn them in this section.

2. Linear Functions Equations that can be written f(x) = mx + b slope
y-intercept
The domain of these functions is all real numbers.

3. Constant Functions f(x) = b, where b is a real number
Would constant functions be even or odd or neither?
The domain of these functions is all real numbers.
The range will only be b

4. Identity Function f(x) = x f(x) = x, slope 1, y-intercept = 0
If you put any real number in this function, you get the same real number “back”.
f(x) = x
Identity Function
f(x) = x, slope 1, y-intercept = 0
Would the identity function be even or odd or neither?
Continuous
Increasing (-∞, ∞)
The domain of this function is all real numbers.
The range is also all real numbers

5. Square Function f(x) = x2
Would the square function be even or odd or neither?
Continuous
Bounded below
Increasing [0, ∞]
Decreasing (-∞ , 0]
The domain of this function is all real numbers.
The range is all NON-NEGATIVE real numbers

6. Cube Function
f(x) = x3
Would the cube function be even or odd or neither?
Continuous
Increasing for all x
Not bounded above or below
No local extrema
No Horizontal asymptotes
No vertical asymptotes
End behavior: As x approaches -∞ the function = -∞. As x approaches ∞ the function = ∞
The domain of this function is all real numbers.
The range is all real numbers

7. Square Root Function
Would the square root function be even or odd or neither?
Continuous on [0, ∞]
Increasing on [0, ∞]
Bounded below but not above
Local minimum at x = 0
No horizontal asymptotes
No vertical asymptotes
End behavior: As x approaches ∞ the square root function = ∞
The domain of this function is NON-NEGATIVE real numbers.
The range is NON-NEGATIVE real numbers

8. Reciprocal Function
The domain of this function is all NON-ZERO real numbers.
Would the reciprocal function be even or odd or neither?
Not continuous, has an infinite discontinuity at x = 0
The range is all NON-ZERO real numbers.

9. Absolute Value Function
The domain of this function is all real numbers.
Would the absolute value function be even or odd or neither?
Continuous
Decreasing on (∞, 0]; increasing on [0, ∞)
Bounded below
Local minimum at (0, 0)
No horizontal asymptotes
No vertical asymptotes
End behavior: As x approaches -∞ |x| = ∞ and as x approaches ∞ |x| = ∞
The range is all NON-NEGATIVE real numbers

10. Exponential Function The range of this function is {y| y > 0 }
f(x) =
Would the exponential function be even or odd or neither?
Exponential Function
The domain of this function is (-∞, ∞)
Continuous
Increasing for all x
Bounded below, but not above
No local extrema
Horizontal asymptote: y = 0
No vertical asymptotes
End behavior: As x approaches -∞ e^x = 0 As x approaches ∞ e^x = ∞
The range of this function is {y| y > 0 }

11. f(x) = ln x Logarithmic Graph
Would the absolute value function be even or odd or neither?
Logarithmic Graph
The domain of this function is {x| x > 0}
Continuous on (0, ∞)
Increasing on (0, ∞)
No symetry
Not bounded above or below
No local extrema
No horizontal asymptotes
Vertical asymptote: x = 0
End behavior: As x approaches ∞ ln x = ∞
The range of this function is (-∞,∞)

12. Sine Function The domain of this function is all real numbers
f(x) = sin x
Would the sine function be even or odd or neither?
Sine Function
The domain of this function is all real numbers
Continuous
Alternately increasing and decreasing in periodic waves
Symmetric with respect to the origin (odd)
Bounded
Absolute maximum of 1
Absolute minimum of -1
No horizontal asymptotes
No vertical asymptotes
End behavior: as the limit of x approaches -∞ of sin x and the limit of x approaches ∞ sinx the limit does not exist
The range of this function is [ -1, 1]

13. Cosine Function f(x) = cos x
Would the cosine function be even or odd or neither?
Cosine Function
Continuous
Alternately increasing and decreasing in periodic waves
Bounded
Absolute maximum 1
Absolute minimum of -1
No Horizontal asymptotes
No Vertical asymptotes
End behavior: As the limit approaches -∞ and the limit approaches ∞ cos x do not exist. (The function values continually oscillate between -1 and 1 and approach no limit.)
The domain of the function is all real numbers
The range of this function is [-1, 1]

14. The Greatest integer Function
f(x) = int (x)
Would the greatest integer function be even or odd or neither?
The Greatest integer Function
Discontinuity at every integer value of x. It is not a continuous function
Increasing/Decreasing behavior: constant on intervals of the form [k, k+1], where k is an integer;
No boundedness
Local extrema: every non-integer is both a local minimum and local maximum;
Horizontal asymptotes: none
Vertical asymptotes: none;
End behavior: int (x) = -∞ As x approaches -∞ and int(x) = ∞ As x approaches ∞.
The domain of this function is all real numbers
The range of this function is all integers

15. The Logistic Function The domain of this function is all real numbers
f(x) =
Would the logistic function be even or odd or neither?
The Logistic Function
The domain of this function is all real numbers
Continuous
Increasing for all x
Bounded below and above
Symmetric about (0, ½), but neither even nor odd
No local extrema
Horizontal asymptotes: y = 0 and y = 1
No vertical asymptotes
End behavior: as x approaches -∞ f(x) = 0 and as x approaches ∞ f(x) = 1
The range of this function is 0 < y <1

16. These are functions that are defined differently on different parts of the domain.
WISE
FUNCTIONS

17. This then is the graph for the piecewise function given above.
This means for x’s less than 0, put them in f(x) = -x but for x’s greater than or equal to 0, put them in f(x) = x2
What does the graph of f(x) = -x look like?
What does the graph of f(x) = x2 look like?
Remember y = f(x) so let’s graph y = - x which is a line of slope –1 and y-intercept 0.
Remember y = f(x) so lets graph y = x2 which is a square function (parabola)
Since we are only supposed to graph this for x< 0, we’ll stop the graph at x = 0.
Since we are only supposed to graph this for x  0, we’ll only keep the right half of the graph.
This then is the graph for the piecewise function given above.

18. For x > 0 the function is supposed to be along the line y = - 5x.
For x = 0 the function value is supposed to be –3 so plot the point (0, -3)
For x values between –3 and 0 graph the line y = 2x + 5.
Since you know the graph is a piece of a line, you can just plug in each end value to get the endpoints. f(-3) = -1 and f(0) = 5
Since you know this graph is a piece of a line, you can just plug in 0 to see where to start the line and then count a – 5 slope.
open dot since not "or equal to"
So this the graph of the piecewise function
solid dot for "or equal to"