Unit 1

Functions

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Part 1.4A corresponds to section 7.1, starting on page 334 of your Pre-Calculus 12 textbook.

So far in this Unit, you have learned about various types of functions and equations and their corresponding graphs.

Each category of function has its own set of characteristics. The same is true of exponential functions.

In an exponential function or equation, there is a power with a numerical base raised to a variable exponent. An example of an exponential function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math». The numerical base is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» and the variable is in the exponent.

A general exponential function is of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»c«/mi»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»c«/mi»«/mstyle»«/math» is the numerical base. The value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»c«/mi»«/mstyle»«/math» must be greater than zero and it cannot be equal to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math».

Why must «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»c«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»c«/mi»«mo»§#8800;«/mo»«mn»1«/mn»«/mstyle»«/math»?

If the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»c«/mi»«/mstyle»«/math» was «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math», the function would always equal zero. Zero to any exponent is just zero multiplied by itself. As such, the function would simplify to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mn»0«/mn»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»0«/mn»«/mstyle»«/math», which is a constant function, not an exponential one.

If the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»c«/mi»«/mstyle»«/math» was «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math», the numerical base of the exponential function would be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math». But, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» to any exponent is just «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math» multiplied by itself. As such, the function would simplify to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mn»1«/mn»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»1«/mn»«/mstyle»«/math», which is a constant function, not an exponential one.

The base, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»c«/mi»«/mstyle»«/math», must also be greater than zero. Recall powers with rational exponents can be thought of as radicals. The radicand of radical functions with even indices must always be positive. As such, the base of an exponential function must always be positive. For example, given «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mi»x«/mi»«/msup»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mo»=«/mo»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«mo»=«/mo»«msqrt»«mo»§#8722;«/mo»«mn»2«/mn»«/msqrt»«/mrow»«/mstyle»«/math», which is not real.

Some more involved examples of exponential functions are as follows.


The domain of an unrestricted exponential function includes all real numbers.