Library of Functions

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

2. Odd and Even Functions A function f is even if f(x) = f(x) for all x in its domain. Vertical line of reflection at x = 0 A function f is odd if f( x) =  f(x) for all x in its domain. Point of rotational symmetry

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

4. f(x) = 3 f(x) = -1 f(x) = 1 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

5. 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? The domain of this function is all real numbers. The range is also all real numbers

6. Square Function f(x) = x2 Would the square function be even or odd or neither? The domain of this function is all real numbers. The range is

7. Cubic Function f(x) = x3 Would the cube function be even or odd or neither? The domain of this function is all real numbers. The range is all real numbers

8. Square Root Function Would the square root function be even or odd or neither? The domain of this function is The range is

9. Reciprocal Function The domain of this function is all NON-ZERO real numbers. Would the reciprocal function be even or odd or neither? The range is

10. Absolute Value Function The domain of this function is all real numbers. Would the absolute value function be even or odd or neither? The range is

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

12. 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.

13. 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"

14. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au