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Mathematical Induction and Evaluating Functions

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1. Prove, using induction, that (1 + x)^n > 1 + nx for n = 2,3,4... if x > -1, x /= 0.

2. If f(x) = (3x +1)/(x - 2), x /= 2
find:

a) [5f(-1) - 2f(0) + 3f(5)] / 6
b) [f(-1/2)]^2
c) f(2x - 3)
d) f(x) + f(4/x), x /= 0
e) [f(h) - f(0)]/h, h /= 0
f) f[f(x)]

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Solution Summary

This solution calculates whether the proposition holds for (1) as well as the solutions for (2) formatted in the attached Word document. This solution provides all of the required steps and calculations.

Solution Preview

1. For x = 2, P(1): (1 + x)2 = 1 + x2 + 2x > 1 + 2x.
Therefore, the proposition holds for n = 2.
Suppose that the proposition ...

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