Properties of Functions

1. 10 5 (0,0) 5 10 (2,10) Properties of Functions (-2 6) (5,0) (-8,-4) There are local minima at f(0)= 0 and f(5)= 0 The local minima are (0, 0) and (5,0) There are local maxima at f(-2) =6 and f(2) =10 The local maxima are (-2,6) and (2,10) f is increasing on the intervals(-8,-2), (0,10), f is decreasing on the intervals (-2,0) (2,5)

2. Library of Functions Linear Function The graph of a linear function is a non-vertical line. A linear function is of the form y = f(x) = mx + bwhere m and b are real numbers. Here m is the slope and b is the y - intercept. To find the x - intercept let y = 0 and solve for x The domain and range of a linear function are all real numbers. Graph

3. Example • f(x) = 2x+ 3 • The slope is m = 2 • The y –intercept is 3 • The x – intercept is –3/2 • Since the slope is positive, f is an increasing function • The domain and the range are the set of real numbers

4. Example f (x) = 4 - 7x The slope is m= -7 The y –intercept is 4 The x – intercept is 7/4 Since the slope is negative, f is an decreasing function The domain and the range are all real numbers

5. The constant function f (x) = C The domain of the constant function is all real numbers The range is the constant C. In this function is equal to 3 The graph is a horizontal line The slope is m = 0 The y – intercept is (0, 3)

6. Identity Function f(x) = x f is called the identity function because the value of y is always identical to that of x f is a linear function because its graph is a straight line. The domain and range are all real numbers The x and y intercepts are both 0 The identity function divides the first and third quadrants into 45 degree angles

7. The Square function The domain of the square function is all real numbers The range of f(x) is The x and y intercepts are (0,0) The square function is even. Thus it is symmetric with the origin

8. The Cube Function The domain and the range are all real numbers The x and y intercepts are at (0,0) The cube function is an odd function and thus is symmetric with the origin

9. The square root function The domain is The range is

10. The Cube root function The domain and range are all real numbers The cube root function is an odd function and is symmetric with the origin

11. The Absolute Value Function • The domain of the absolute value function is all real numbers. • The range is all non negative numbers • The absolute value function is even and thus symmetric with the origin

12. The Reciprocal Function • The domain is all real numbers except x = 0 • The range is all real numbers except y= 0 • The reciprocal function is an odd function and is thus symmetric with the origin