Answer to Question #144456 in Algebra for Ankita Sonawane

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

First g(x) = h(x):

2x - 5 = 8/x + 10

2x - 8/x - 15 = 0

x not equal 0:

2x² - 15x - 8 = 0

Discriminant = (-15)² - 4 * (-8) *2 = 289 = 17²

So, roots of the equation: x₁ = (15 + 17) / (2 * 2) = 8

x₂ = (15 - 17) / (2 * 2) = -0.5

Now let's analyze f(x):

1) f(8) = 2^{(8 - 5)} + 3 = 2³ + 3 = 11 equal g(8) = 2 * 8 - 5 = 11

2) f(-0.5) = 2^{(-0.5 - 5)} + 3 = 3.022 not equal g(-0.5) = 2 * (-0.5) - 5 = -6

So the only root is x = 8.