Q34. - Algebra 2

Question

Graph each relation or equation and find the domain and range. 

Then determine whether the relation or equation is a function.

$x=2y^2-3$

Step-by-Step Solution

Answer

The graph of the given equation is:

The domain of the given equation is $\left\{x|x\ge -3\right\}$ and the range of the given equation is all real numbers. The given equation is not a function.

Step 1 – Vertical line test, domain and range.

• The graph represents a function if no vertical line intersects the graph on more than one point. The graph does not represent a function if some vertical line intersects the graph on two or more points.
• The set of input values is known as domain.

The set of output values is known as Range.

Step 2 – Draw the graph of the given relation.

The given equation is $x=2y^2-3$.

The table of values for the given equation is:

$\begin{array}{cc}x& y\\ 5& -2\\ -1& -1\\ -3& 0\\ -1& 1\\ 5& 2\end{array}$

Plot these ordered pairs on a coordinate plane and connect them by a free-hand curve as shown below:

Step 3 – Domain and range of the given equation.

From the above graph it is clear that only real numbers greater than $-3$ are $x$ -coordinates of points on the graph. So, the domain of the given equation is $\left\{x|x\ge -3\right\}$

 

Every real number is the $y$ -coordinate of some point on the graph, so the range is all real numbers.

Step 4 – Check whether the equation is a function.

The above table and graph it is clear that there are two $y$ values for each possible $x$ -value except $-3$. So, the graph does not pass the vertical line test and the given equation is not a function.

 

Thus, the domain of the given equation is $\left\{x|x\ge -3\right\}$ and the range of the given equation is all real numbers. The given equation is not a function.