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HIGH SCHOOL DIFFERENTIAL CALCULUS COURSE 1.3 SUPPORT 1 – ABSOLUTE VALUE FUNCTIONS 1.3.1. Define the absolute value function. The absolute value or modulus of a real number is its numerical value without its sign. The notation of absolute value of a value was introduced first by Karl Weierstrass in 1841. The absolute value of a negative number makes it positive. Because of how the absolute value function behaves, it is important to include also negative inputs in your T-chart when graphing the function. The absolute value function is defined as f ( x) x , f : , 0, which means that the function is taking any real number for its domain and is getting only zero or positive numbers as range. (Remember: f : Domain Range ) For example, if we want to graph this function f ( x) x , a general error could be made if we use only positive x values in the table of values. X f ( x) x 0 0 1 1 2 2 3 3 Using only this table of values, the function could be mistakenly graphed. y y x Plotting the points with this table of values, we get a wrong graph for the function x Be careful, when you work with absolute value functions, always give values in the negative and positive side. So: X f(x) We plot the new points and we get: -3 3 -2 2 -1 1 0 0 1 1 2 2 3 3 y y This is the graph of f ( x) x x x 1.3.2. Explore the effect of the absolute value when it is applied over any type of function use of a graphing utility. by the If we are trying to apply the absolute value of other functions, the effect is the same. Domain Domain y x , y x , x x Range Range y 3, y 0, y x2 3 y x2 3 y y Domain x 0, Domain x 0, x x Range Range y , y 0, y ln x y ln x y y Domain x , Domain x , x x Range y 1, 1 y sin x Range y 0, 1 y sin x From these functions we can observ that the absolute-value will flip the negative part of the graph up into the positive values (above the x-axis). 1.3.3. Graph the following absolute value functions, with absolute value. 1.3.4. Determine the domain and range of a function with absolute value. Given the function f ( x) x x 2 2 To graph the function f ( x) x 2 x 2 we have to decide the vertex first. Completing the square, we find 2 1 9 1 9 that the function it can be written as f ( x ) x so the vertex is V , . 2 4 2 4 Using the transformations of functions, the basic function f ( x) x 2 is moved 1 9 to the right and 2 4 downwards. Graphing the quadratic function , we get: y y x Applying the absolute value of this function, the negative side of the function, below the x axis, is reflected about the x axis x axis of symmetry axis of symmetry The domain of this function is: The domain of this function is: x , x , The range is: The range is: 9 y , 4 y 0,