Cambridge STEP Mathematics Question Review (2024-08-24)

1. Denote 2 exponential functions:

f(x) = 2^x and g(x) = 3^x

(a) Show that they are bijective.

(b) If f(x)=g(x), show that they have only 1 real solution.

Solution:

1. f"(x) > 0 and g"(x) > 0, and both f(x) and g(x) has no local extrema for all real x. Given they have real inverse functions over the line of symmetry y=x, they must be bijective.

2. Put x=0, you are done.

Since exponential functions are not commutative, they will intersext only once.

intersect*

Bonus Question:

Given f(x) = x^5 + x^4 + x^3 + x^2 + 2i =0,

show that the product of roots are not commutative.