04Gabsvalues


 * Absolute Values and Exponential Graphs **

Recall the absolute value function, y = |f(x)| (also called the ** modulus function **).

It causes all negative y-values of y = f(x) to be reflected above the x-axis.

As a hybrid function this is written as: math y = \big| f(x) \big| = \left\{ \begin{matrix} +f(x) \, \text{ for } f(x) \geqslant 0 \\ -f(x) \, \text{ for } f(x) < 0 \\ \end{matrix} \right. math

Also the composite function, y = f(|x|), causes the left side of the graph {left of the y-axis} to be a reflection across the y-axis of the right side of the graph.

math y = f \big( |x| \big) = \left\{ \begin{matrix} f(+x) \, \text{ for } x \geqslant 0 \\ f(-x) \, \text{ for } x < 0 \\ \end{matrix} \right. math

These concepts can be applied to exponential graphs and logarithmic graphs.

Graph of y = ln|x|

A very common composite is the graph of y = log e |x|. It is a combination of the graph of y = ln(x) and y = ln(–x). In other words, it is the graph of y = ln(x) and a reflection of y = ln(x) across the y-axis.



Domain: x Î R\{0}

Range: y Î R

Asymptote: x = 0

x-intercepts: (–1, 0) (1, 0)

No y-intercepts


 * Example 1 **

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 * 1) Find the exact values of the intercepts of the following two graphs
 * 2) Sketch y = |log e (x + 2)| – 1 and state domain, range, intercepts(2 decimal places) and asymptotes
 * 3) Sketch y = log e (|x| + 2) – 1 and state domain, range, intercepts (2 decimal places) and asymptote

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