A function is a relation in which each input value corresponds to exactly one output value.
B
b
Solve x=y^2 for y or compare the graph of y=sqrt(x) to the graph of x=y^2.
C
c
Use the information from Part B. Note that the lower part of the graph is a reflection of the upper part of the graph.
A
a
No, see solution.
B
b
See solution.
C
c
y=- sqrt(x), see solution.
Practice makes perfect
a
We are asked to determine if the given graph of x=y^2 is a graph of a function.
In order to do that, recall that a function is a relation in which each input value corresponds to exactly one output value. In our case, we need to check that each x-value corresponds to no more than one y-value. We can check this using the Vertical Line Test.
As we can see, all but one x-value, the point at (0,0), corresponds to two y-values. Therefore, the given graph does not illustrate a function.
b
To determine the relation between x=y^2 and the square root function y=sqrt(x), let's solve the first of these equations for y.
Using the properties of absolute values, this equation can be split into two equations.
lc y ≥ 0:y = sqrt(x) & (I) y < 0:y = - sqrt(x) & (II)
As we can see, the Equation (I) is the same equation as the square root function that we were given. Therefore, y=sqrt(x) is the upper half of the relation described by x=y^2.
c
Let's start by identifying which part of the given graph is shown in Quadrant IV.
In Part B, we found that the equation x=y^2 can be written as the following two equations.
x=y^2 ⇒ lcy=sqrt(x) & (I) y=- sqrt(x) & (II)
The graph of the Equation (I) is the upper half of the graph of x=y^2 and it is shown in Quadrant I. Therefore, the lower part of the graph — shown in Quadrant IV — is the graph of the Equation (II).
To double check this conclusion, we can graph y=- sqrt(x) by making a table of values.
x
0
1
2
4
9
Substitute
y=- sqrt(0)
y=- sqrt(1)
y=- sqrt(2)
y=- sqrt(4)
y=- sqrt(9)
y
0
- 1
- 1.4
- 2
- 3
Now, we can plot the points and connect them with a smooth curve.
Another possible way to graph this function is by reflecting the graph of y=sqrt(x) over the x-axis. Either way, we get the exact same graph as the part of x=y^2 that is shown in Quadrant IV.
