Unit 1

Functions

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Part 1.1A corresponds to section 2.1, Radical Functions and Transformations, starting on page 60 of your Pre-Calculus 12 textbook.

A radical operation includes some type of root. Below is a list of common roots.



Let us look at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mn»4«/mn»«mn»2«/mn»«/msup»«mo»=«/mo»«mn»16«/mn»«/mrow»«/mstyle»«/math» in detail.

Notice that taking the square root of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»16«/mn»«/mstyle»«/math» takes you back to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math».


Squaring and taking the square root are opposite operations, as discussed in the table above.

You may recall there are two different values that can be squared to give «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»16«/mn»«/mstyle»«/math».


So, consider «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msqrt»«mn»16«/mn»«/msqrt»«/mstyle»«/math». Is this equivalent to a «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math» or a «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«mn»4«/mn»«/mstyle»«/math»? By convention, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msqrt»«mn»16«/mn»«/msqrt»«/mstyle»«/math» implies the positive square root and is called the principal square root. Also, note that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/msqrt»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math» only if «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».

Recall that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»1«/mn»«/msup»«/mrow»«/mstyle»«/math». Any value can be written as a number to the power of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math». Now, think about exponent laws. If a power is raised to a power, the exponents are multiplied.


Now, apply the exponent laws to determine the opposite exponent to squaring a number. Raise «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mstyle»«/math» to the power of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» such that the result is the original value, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» (or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mi»x«/mi»«mn»1«/mn»«/msup»«/mstyle»«/math»).


Multiply the powers.


Now, solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».


So, when using positive values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mfenced»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«mo»=«/mo»«mi»x«/mi»«/mrow»«/mstyle»«/math». In other words, the power of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/math» is the same as the square root.


Likewise, other radicals can be expressed using rational exponents.


Notice the denominator of the rational exponent is the index of the radical.

Radicals with even indices—square roots, fourth roots, etc.—must have a radicand that is greater than or equal to zero. Radicals with odd indices—cube roots, fifth roots, etc.—may have a radicand that is positive, negative, or zero.

Because of their equivalence, both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mi»x«/mi»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/msup»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mroot»«mi»x«/mi»«mn»3«/mn»«/mroot»«/mstyle»«/math» are considered radicals. As such, a radical function can be expressed using either notation, so long as there is a variable in the radicand (or as the base in a power).