function notation and transformation of functions

bumblebee123

Junior Member
can anyone help me to understand the answer to this question?

question: the function f is defined as f (x) = (x + 3) / (x - 1 )

any help would be really appreciated

ksdhart2

Senior Member
One would naturally presume that $$ff(x)$$ is function composition notation and means the same thing as $$f(f(x))$$. That is to say, it's asking you to evaluate the function $$f(x)$$ with $$f(x)$$ passed as an argument. So just write out the given definition of $$f(x)$$, replacing each $$x$$ with $$f(x)$$ and then in turn replace $$f(x)$$ with its definition again:

$$\displaystyle f(f(x)) = \frac{f(x)+3}{f(x)-1} = \text{???}$$

pka

Elite Member
can anyone help me to understand the answer to this question?
question: the function f is defined as f (x) = (x + 3) / (x - 1 ) find ff(x). give your answer in its simplest form.
I have a question. Was the question actually given as $$\displaystyle f\,f(x)~?$$
OR was it written as either $$\displaystyle f\circ f(x)$$ or $$\displaystyle f\cdot f(x)~?$$

bumblebee123

Junior Member
I have a question. Was the question actually given as $$\displaystyle f\,f(x)~?$$
OR was it written as either $$\displaystyle f\circ f(x)$$ or $$\displaystyle f\cdot f(x)~?$$
it was given as ff( x )

bumblebee123

Junior Member
One would naturally presume that $$ff(x)$$ is function composition notation and means the same thing as $$f(f(x))$$. That is to say, it's asking you to evaluate the function $$f(x)$$ with $$f(x)$$ passed as an argument. So just write out the given definition of $$f(x)$$, replacing each $$x$$ with $$f(x)$$ and then in turn replace $$f(x)$$ with its definition again:

$$\displaystyle f(f(x)) = \frac{f(x)+3}{f(x)-1} = \text{???}$$
so f ( f (x) ) = ( ( ( x+3) / ( x -1 )) + 3 ) / ( ((x +3 ) / ( x - 1 ) )) - 1)

HallsofIvy

Elite Member
Yes, now do the algebra to simplify that.

bumblebee123

Junior Member
Yes, now do the algebra to simplify that.
( 4x / (x-1) ) / ( 4 / (x-1) ) = ( 4x^2 - 4x) / ( 4x - 4) = ( 4 ( x^2 - x ) ) / ( 4 ( x-1) ) = (x^2 - x ) / ( x -1 ) = ( x ( x - 1) ) / ( x -1 ) = x

so ff(x) = x which is correct

thanks

pka

Elite Member
\displaystyle \begin{align*}\frac{{\frac{{x + 3}}{{x - 1}} + 3}}{{\frac{{x + 3}}{{x - 1}} - 1}} &= \frac{{\left( {\frac{{x + 3}}{{x - 1}} + 3} \right)(x - 1)}}{{\left( {\frac{{x + 3}}{{x - 1}} - 1} \right)(x - 1)}}\\& = \frac{{x + 3 + 3(x - 1)}}{{x + 3 - (x - 1)}}\\& = \frac{{4x}}{4}\\& = x \end{align*}