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

Firstly, let us find all points where the functions $g(x)=2x-5$ and $h(x)=\frac{8}{x}+10$ have the same value. For this solve the following equation:

$2x-5=\frac{8}{x}+10$

which is equivalent to

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

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

$2x^2-16x+x-8=0$

$2x(x-8)+x-8=0$

$(2x+1)(x-8)=0$

$x=8$ or $x=-\frac{1}{2}$

Since $f(8)=2^3+3=11$ and $g(8)=h(8)=11$, at point $(8,11)$ the functions $f(x),g(x),h(x)$ have the same value.

Since $f(-\frac{1}{2})=2^{-\frac{1}{2}-5}+3>0$ and $g(-\frac{1}{2})=-6<0$, in this case the functions $f(x),g(x),h(x)$ have different values.

Answer: $(8,11)$