Answer to Question #192818 in Algebra for Mal

Question

If f(x)=2x+5 , then the graphs of y=f(x) and y=squareroot f(x) will have two common points.

a. What are the y-coordinates of these common points?

b. Determine the x-coordinates of these common points, and then state the points.

Answer 1

given

$f(x)=2x+5$

$let\space g(x)=\sqrt{\smash[b]{f(x)}}=\sqrt{\smash[b]{2x+5}}$

The graphs of $y=f(x)$ and $y=g(x)$ will intersect at points where $f(x)=g(x)$

$f(x)=g(x)\Rightarrow 2x+5=\sqrt{\smash[b]{2x+5}}$

$\Rightarrow2x+5-\sqrt{\smash[b]{2x+5}}=0$

$\Rightarrow\sqrt{\smash[b]{2x+5}}(\sqrt{\smash[b]{2x+5}}-1)=0$

$\Rightarrow\sqrt{\smash[b]{2x+5}}=0,\sqrt{\smash[b]{2x+5}}=1$

$\Rightarrow2x+5=0,2x+5=1$

$\Rightarrow x=-\frac{5}{2} \space ,\space x=-2$

The y-coordinates are

$x=-2\Rightarrow y=\sqrt{\smash[b]{2x+5}}=1$

$x=-\frac{5}{2}\Rightarrow y=\sqrt{\smash[b]{2x+5}}=0$

Hence, the x-coordinates of the points of intersection are $x=-\frac{5}{2},-2$

Hence, the x-coordinates of the points of intersection are $y=0,1$

The points of intersections are $(-2,1)$ and $(-\frac{5}{2},0)$