Q. 20 - Calculus

Question

Match the two graphs shown here to the equations  $x{y}^2+x=1$ and  $x{y}^2+x+1=0$Explain your choices.

Step-by-Step Solution

Answer

The graph of the function:$x{y}^2+x=1$:

Graph of $x{y}^2+x+1=0$

Step 1. Given information:

The functions are:

$$\begin{gathered} x{y}^2+x=1 \\ 1+x+x{y}^2=0 \end{gathered}$$

Step 2. Find x - intercepts of both functions.

Substitute $y=0$ in both functions and solve for $x$.

$$\begin{gathered} x{y}^2+x=1 \\ x(1+y^2)=1 \\ x(1+0)=1 \\ x=1 \end{gathered}$$

So x-intercept of this function is $(1,0)$,

$$\begin{gathered} 1+x+x{y}^2=0 \\ 1+x+x.0=0 \\ 1+x=0 \\ x=-1 \end{gathered}$$

$x-$intercept is $(-1,0)$

Step 3. Match the graphs with their function.

Since $x{y}^2+x=1$, has $x-$intercept is $(1,0)$,so the graph will cross at this point.

The $x-$intercept of the function $1+x+x{y}^2=0$ is $(-1,0)$,so the graph will cross $x-$axis at this point.