Question

Find all points (x, y) where the functions f(x), g(x), h(x) have the same value:

Answer 1

Answer:

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Solve them pairwise:

2^(x-5) + 3 = 2x - 5, so

2^(x-5) = 2x - 8, so

(2^x)/32 = 2x - 8, so

2^x = 64x - 128, so

x·log(2) = log(64x - 128), so

x = log(64x - 128) / log(2) = log₂(64x - 128).

2x - 5 = 8/x + 10, so

2x = 8/x + 15, so

x = 4/x + 15/2, so

x² = 4 + (15/2)x, so

x² - (15/2)x - 4 = 0, so

x = 8 or -1/2.

Substitute:

x=8:

2x - 5 → 2(8) - 5 = 11, so (8,11) is a possible point of intersection.

x=-1/2:

2x - 5 → (-1/2) - 5 = -6, so (-1/2,-6) is possible.

x=8:

8x + 10 → 8/8 + 10 = 11, so (8,11) is again possible.

x=-1/2:

8/x + 10 → 8/(-1/2) + 10 = -6, so (-1/2,-6) is again possible.

x=8:

2^(x-5) + 3 → 2^(8/5) + 3 = 8+3 = 11, so (8,11) is again possible.

x=-1/2:

2^(x-5) + 3 → 2^(-1/2 - 10/2) + 3 ≈ 3.022, so we get (-1/2,3.022).

This point does not match the other points that use x=-1/2, but all three equations have the same value (11) when x=8.

The solution point is then (8,11).