Q33. - Algebra 2

Question

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

Then determine whether the relation or equation is a function.

$y=x^2$

Step-by-Step Solution

Answer

The graph of the given equation is:

The domain of the given equation is all real numbers and the range of the given equation is $\left\{y|y\ge 0\right\}$. The given equation is 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 $y=x^2$.

The table of values for the given equation is:

$\begin{array}{cc}x& y\\ -2& 4\\ -1& 1\\ 0& 0\\ 1& 1\\ 2& 4\end{array}$

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 every real number is the $x$-coordinate of some point on the graph. So, the domain of the given equation is all real numbers

 

Only real numbers greater than $0$ are $y$-coordinates of points on the graph, so the range is $\left\{y|y\ge 0\right\}$.

Step 4 – Check whether the equation is a function.

The above graph passes the vertical line test. For each $x$ value, there is exactly one $y$ value. So, the given equation is a function.

 

Thus, the domain of the given equation is all real numbers and the range of the given equation is $\left\{y|y\ge 0\right\}$. The given equation is a function.