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

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Presentation transcript:

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

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

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

Constant Functions f(x) = b, where b is a real number The domain of these functions is all real numbers. The range will only be b f(x) = 3f(x) = -1f(x) = 1 Would constant functions be even or odd or neither?

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

Square Function f(x) = x 2 The domain of this function is all real numbers. The range is Would the square function be even or odd or neither?

Cubic Function f(x) = x 3 The domain of this function is all real numbers. The range is all real numbers Would the cube function be even or odd or neither?

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

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

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

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

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) = x 2 What does the graph of f(x) = -x look like? Remember y = f(x) so let’s graph y = - x which is a line of slope –1 and y-intercept 0. Since we are only supposed to graph this for x< 0, we’ll stop the graph at x = 0. What does the graph of f(x) = x 2 look like? Since we are only supposed to graph this for x  0, we’ll only keep the right half of the graph. Remember y = f(x) so lets graph y = x 2 which is a square function (parabola) This then is the graph for the piecewise function given above.

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 For x = 0 the function value is supposed to be –3 so plot the point (0, -3) For x > 0 the function is supposed to be along the line y = - 5x. 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. So this the graph of the piecewise function solid dot for "or equal to" open dot since not "or equal to"

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