Question

Find all points (x, y) where the functions f(x), g(x), h(x) have the same value:
f(x) = 2^x−5 + 3, g(x) = 2x − 5, h(x) = 8/x+ 10

Answer 1

$f(x)=2^{x-5}+3,\ \ g(x)=2x-5, \ \ h(x)=8/x+10$ .

Let us find all points $(x,y)$ where functions $g(x)$ and $h(x)$ have the same value.

So, $g(x)=h(x)=y$ .

$2x-5=8/x+10$ (multiply both sides of equation by $x$ )

$2x^2-5x=8+10x$

$2x^2-15x-8=0$

$x_{1,2}=\frac{15\pm \sqrt{15^2-4\times 2\times (-8)}}{2\times 2}=\frac{15\pm17}{4}$

$x_1=8,\ \ x_2=-1/2$

Now we calculate $y_1=g(x_1)=11$ , $y_2=g(x_2)=-6$ .

We have two points $(8,\ 11)$ and $(-1/2,\ -6)$ where functions $g(x)$ and $h(x)$ have the same value.

Let is find which of these points satisfy the condition, that functions $f(x)$ , $g(x)$ and $h(x)$ have the same value.

$f(x_1)=2^3+3=11=y_1$

$f(x_2)=2^{-1/2-5}+3=\frac{1}{32\sqrt{2}}+3\cancel{=}-6=y_2$ .

Answer: $(8,\ 11)$ .