The Absolute Value Function and Piecewise Functions
Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. For example, consider the piecewise definition of the absolute value function:
The piecewise definition of the absolute value function
For all values of x less than zero, the first function (−x) is used, which negates the sign of the input value, making negative numbers positive. For all values of x greater than or equal to zero, the second function (x) is used, which evaluates trivially to the input value itself.
More references and links to graphing, graphs and absolute value functions.
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Example: f is a function given by
f (x) = |(x - 2^)2 - 4|
Find the x and y intercepts of the graph of f.
Find the domain and range of f.
Sketch the graph of f.
Solution to Example 2
a - The y intercept is given by
(0 , f(0)) = (0 ,(-2)^2 - 4) = (0 , 0)
The x coordinates of the x intercepts are equal to the solutions of the equation
|(x - 2)^2 - 4| = 0
which is solved
(x - 2)^2 = 4
Which gives the solutions
x = 0 and x = 4
The x intercepts is at the point (0 , 0) and (4 , 0)
b - The domain of f is the set of all real numbers
Since |(x - 2)^2 - 4| is either positive or zero for x = 4 and x = 0; the range of f is given by the interval [0 , +infinity).
c - To sketch the graph of f(x) = |(x - 2)2 - 4|, we first sketch the graph of y = (x - 2)2 - 4 and then take the absolute value of y.
The graph of y = (x - 2)^2 - 4 is a parabola with vertex at (2,-4), x intercepts (0 , 0) and (4 , 0) and a y intercept (0 , 0). (see graph below)
graph of y = (x - 2)<sup>2</sup> - 4
The graph of f is given by reflecting on the x axis part of the graph of y = (x - 2)^2 - 4 for which y is negative. (see graph below).
graph of y = |(x - 2)<sup>2</sup> - 4|
This Shows you how to solve a Quadratic with absolute values
Graphing
a harder example (I need to check f this could come up - Mr Green)
another example like above (I need to check f this could come up - Mr Green)
The Absolute Value Function and Piecewise Functions
Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains. For example, consider the piecewise definition of the absolute value function:
For all values of x less than zero, the first function (−x) is used, which negates the sign of the input value, making negative numbers positive. For all values of x greater than or equal to zero, the second function (x) is used, which evaluates trivially to the input value itself.
More references and links to graphing, graphs and absolute value functions.
Is this worth having??
Example: f is a function given by
f (x) = |(x - 2^)2 - 4|
Solution to Example 2
(0 , f(0)) = (0 ,(-2)^2 - 4) = (0 , 0)
|(x - 2)^2 - 4| = 0
which is solved
(x - 2)^2 = 4
Which gives the solutions
x = 0 and x = 4
Since |(x - 2)^2 - 4| is either positive or zero for x = 4 and x = 0; the range of f is given by the interval [0 , +infinity).
The graph of y = (x - 2)^2 - 4 is a parabola with vertex at (2,-4), x intercepts (0 , 0) and (4 , 0) and a y intercept (0 , 0). (see graph below)
This Shows you how to solve a Quadratic with absolute values
Graphing
a harder example (I need to check f this could come up - Mr Green)
another example like above (I need to check f this could come up - Mr Green)