Special Case: Cube Function x 3
Notes
3
The cube function f (x) = x is both one-to-one and onto. It is
therefore invertible.
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The inverse function is f −1 (x) =
√
3
x.
√
The composition of f and its inverse f −1 looks like ( 3 x)3 = x.
Similarly the composition in the other order looks like
√
3
x 3 = x.
The domain and range of f is R. Therefore the domain and
range for the inverse is also R.
You can take the cube root of any real number.
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Special Case: Square Function x 2
Notes
2
Recall the g (x) = x was not one-to-one nor onto. This can be
modified by limiting the domain and target set.
Suppose h : [0, ∞) → [0, ∞) is defined by h(x) = x 2 .
√
2
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The inverse function is h−1 (x) =
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h and h−1 both have domain and range set equal
to the interval [0, ∞).
You can only take square roots of positive
numbers.
x.
√
√
It is common to use
for 2 . This shorthand only applies to the
square root. For other roots, like the cube root, the number must be
written.
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n-th roots
Notes
If n ∈ N and n ≥ 2, then x n describes a function. We have
summarized:
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When n is an odd exponent, x 3 , x 5 , x 7 , . . . describe one-to-one
and onto functions.
√
When n is odd, n : R → R is the inverse of x n .
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When n is an even exponent, x 2 , x 4 , x 6 , . . . are not one-to-one
and are not onto functions.
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When n is an even exponent, f : [0, ∞) → [0, ∞) where
f (x) = x n is one-to-one and onto. In this case
√
n
: [0, ∞) → [0, ∞) is the inverse of x n .
√
√
n-th roots are inverses of x n . Therefore ( n x)n = x and n x n = x.
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Using n-th roots as inverses
Notes
Recall that when solving an equation, we must keep track of what
operations are happening. To continue, erase what happened last and
write its inverse.
Example 1 Solve for x where 5(x + 1)3 = 20.
• Note which operation is used on the left side. To x, we add one,
then we cube, then we multiply by five. (Go Team PEMDAS!)
Reverse these steps and apply each inverse.
Example 2 Solve for x where
Example 3 Solve for x where
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√
2
√
2
8x = 4.
x + 5 = 11.
Math 1050
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Implied Domains
Notes
Note the odd n-th root functions will always have R as the domain
set.
√
For functions like f (x) = 2 x + 5 and other even n-th root functions,
we will be required to use inequalities to find the implied domain.
Refer to page 87 for a complete list of rules for inequalities.
√
Example 1 What is the implied domain of f (x) = 2 x + 5?
We can check that x + 5 can never be a negative number, so
x + 5 ≥ 0. Then x ≥ −5. This set is [−5, ∞).
Example 2 What is the implied domain of g (x) =
√
2
Example 3 What is the implied domain of h(x) = 7
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6 − 13x?
√
10
6x − 8?
Math 1050
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Expectation: Simplifying Square Roots of Natural Numbers
√
When writing, say 40, we will be expected to write a simplify the
√
result. The result is 2 10. This result does not ask us to take the
square root of a number that is a known square.
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Known squares are the numbers 1, 4, 9, 16, 25, . . .
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It is good practice write out these numbers to recognize them
quickly. (That is, find 12 , 22 , 32 , 42 , . . .)
Recognize them as factors of natural numbers. For instance,
40 = 4 · 10. For that matter notice 10 = 2 · 5, a product of two
prime factors that are not squares.
√ √ √
√
Examples Simplify the expressions 49, 48, 360, 72.
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Notes