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smc

algebra senior

Problem

For all real numbers

x

except

x = 0

and

x = 1

the function

f (x)

is defined by

f (x / (x - 1)) = 1/ x

. Suppose

0 \leq t \leq π /2

. What is the value of

f (sec^{2} t)

?

(A)

s i n^{2} t h e t a

(B)

co s^{2} t h e t a

(C)

t a n^{2} t h e t a

(D)

co t^{2} t h e t a

Solution

Let

y = \frac{x}{x - 1} \Rightarrow x y - y = x \Rightarrow x = \frac{y}{y - 1}

f (y) = \frac{1}{x} = \frac{y - 1}{y} = 1 - \frac{1}{y}

f (se c^{2} t) = s i n^{2} t

s i n^{2} θ

Final answer

A