Finding the intersection of two functions (graphs)

Question

I am trying to solve a problem with functions, but we haven't studied it in school, and I have some problems. I am trying to find the intersection of two functions: $f(x)=\frac{x^2}{x+2}$ and $g(x)=x+1$. I am using GeoGebra, and this is what I have made: So according to the answers, the only one intersection is for $x=-\frac{2}{3}$. How can I see that? Thank you in advance!

Answer 1

In general, the graphs of $y=f(x)$ and $y=g(x)$ intersect when their $y$-values are equal for a common value of $x$. So the $x$-coordinate of the point of intersection must satisfy the equation $$f(x) = g(x)$$ Solving for $x$ tells you what value(s) $x$ must take for the graphs to intersect; then you can verify that they do by evaluating $f$ and $g$ and making sure you get the same value.

So in this case, you need to solve $$\dfrac{x^2}{x+2} = x+1$$ Multiplying by through by $x+2$ and doing some algebra gives you what you want.

Answer 2

$$\begin{align} \frac{x^2}{x+2}=x+1 \quad \iff & x^2=(x+1)(x+2) \\ \iff & x^2=x^2+3x+2 \\ \iff &3x+2=0 \\ \iff &x=-\tfrac{2}{3} \end{align}$$