xx size 12{ sqrt {x} } {} can be defined as the inverse function of
x2x2 size 12{x rSup { size 8{2} } } {}. Recall the definition of an inverse function—
f−1(x)f−1(x) size 12{f rSup { size 8{ - 1} } \( x \) } {} is defined as the inverse of
f1(x)f1(x) size 12{f rSup { size 8{1} } \( x \) } {} if it reverses the inputs and outputs. So we can demonstrate this inverse relationship as follows:
Table 1
| xx size 12{ sqrt {x} } {} is the inverse function of
x2x2 size 12{x rSup { size 8{2} } } {} |
|
3
→
x
2
→
9
3
→
x
2
→
9
size 12{3 rightarrow x rSup { size 8{2} } rightarrow 9} {}
|
|
9
→
x
→
3
9
→
x
→
3
size 12{9 rightarrow sqrt {x} rightarrow 3} {}
|
Similarly,
log2xlog2x size 12{"log" rSub { size 8{2} } x} {} is the inverse function of the exponential function
2x2x size 12{2 rSup { size 8{x} } } {}.
Table 2
| log2xlog2x size 12{"log" rSub { size 8{2} } x} {} is the inverse function of
2x2x size 12{2 rSup { size 8{x} } } {} |
|
3
→
2
x
→
8
3
→
2
x
→
8
size 12{3 rightarrow 2 rSup { size 8{x} } rightarrow 8} {}
|
|
8
→
log
2
x
→
2
8
→
log
2
x
→
2
size 12{8 rightarrow "log" rSub { size 8{2} } x rightarrow 2} {}
|
(You may recall that during the discussion of inverse functions,
2x2x size 12{2 rSup { size 8{x} } } {} was the only function you were given that you could not find the inverse of. Now you know!)
In fact, as we noted in the first chapter,
xx size 12{ sqrt {x} } {}is not a perfect inverse of
x2x2 size 12{x rSup { size 8{2} } } {}, since it does not work for negative numbers.
(−3)2=9(−3)2=9 size 12{ \( - 3 \) rSup { size 8{2} } =9} {}, but
99 size 12{ sqrt {9} } {} is not
−3−3 size 12{ - 3} {}. Logarithms have no such limitation:
log2xlog2x size 12{"log" rSub { size 8{2} } x} {} is a perfect inverse for
2x2x size 12{2 rSup { size 8{x} } } {}.
The inverse of addition is subtraction. The inverse of multiplication is division. Why do exponents have two completely different kinds of inverses, roots and logarithms? Because exponents do not commute.
3232 size 12{3 rSup { size 8{2} } } {} and
2323 size 12{2 rSup { size 8{3} } } {} are not the same number. So the question “what number squared equals 10?” and the question “2 to what power equals 10?” are different questions, which we express as
1010 size 12{ sqrt {"10"} } {} and
log210log210 size 12{"log" rSub { size 8{2} } "10"} {}, respectively, and they have different answers.
x2x2 size 12{x rSup { size 8{2} } } {} and
2x2x size 12{2 rSup { size 8{x} } } {} are not the same function, and they therefore have different inverse functions
xx size 12{ sqrt {x} } {} and
log210log210 size 12{"log" rSub { size 8{2} } "10"} {}.
"This is the "concepts" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about […]"