Extend 4-7: Algebra Lab Drawing Inverses
1. Is the graph of the original relation a function?
Explain.
Placing a vertical line over the graph, you can see that
the line intersects the graph once. This means that the
relation is a function.
2. Is the graph of the inverse relation a function?
Explain.
SOLUTION: SOLUTION: To determine whether the relation is a function, use
the vertical line test.
To determine whether the relation is a function, use
the vertical line test.
Placing a vertical line over the graph, you can see that
the line intersects the graph once. This means that the
relation is a function.
Placing a vertical line over the graph, you can see that the line intersects the graph twice. This means
that the relation is not a function.
2. Is the graph of the inverse relation a function?
Explain.
SOLUTION: To determine whether the relation is a function, use
the vertical line test.
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3. What are the domain and range of the original
relation? of the inverse relation?
SOLUTION: The domain of a relation is the set of values that can
be used for x. In the original relation, x can be any real
number so D = {all real numbers}.
The range of a relation is the set of values that y can
be equal to. In the original relation, y can be any real
number greater than or equal to 0 so R = {y | y ≥ 0}.
For the inverse relation:
The x-values can be any real number greater than or
equal to 0 so D = {x | x ≥ 0}. The y-values can be any
real number so R = {all real numbers}.
4. If the domain of the original relation is restricted to D
= {x | x ≥ 0}, is the inverse relation a function? Explain.
SOLUTION: If the original relation had a domain restricted to {x | x
≥ 0}, the graph would look like half of a parabola that opens up. The inverse would be half of a parabolaPage
that1
opens to the right. Since none of the domain values
would have more than one corresponding range value,
number greater than or equal to 0 so R = {y | y ≥ 0}.
For the inverse relation:
The x-values can be any real number greater than or
equal4-7:
to 0 Algebra
so D = {xLab
| x ≥ 0}. The y-values
Extend
Drawing Inversescan be any
real number so R = {all real numbers}.
4. If the domain of the original relation is restricted to D
= {x | x ≥ 0}, is the inverse relation a function? Explain.
SOLUTION: If the original relation had a domain restricted to {x | x
≥ 0}, the graph would look like half of a parabola that opens up. The inverse would be half of a parabola that
opens to the right. Since none of the domain values
would have more than one corresponding range value,
the inverse relation is a function.
5. If the graph of a relation is a function, what can you
conclude about the graph of its inverse?
SOLUTION: Nothing; its inverse may or may not be a function.
However, if the domain of the original is restricted, the
inverse will likely be a function.
6. CHALLENGE The vertical line test can be used to
determine whether a relation is a function. Write a
rule that can be used to determine whether a function
has an inverse that is also a function.
SOLUTION: Sample answer: Draw a horizontal line to see if the
inverse represents a function. If a horizontal line
intersects the graph more than once, then the inverse
is not a function. If a horizontal line intersects the
graph at only one point, the inverse is a function.
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