SOLUTION: a) For the given functions 𝑓(𝑥)= x/x-1 and g(x)=-4/x ,find the domain of the composite functions 𝑓o𝑔 and 𝑔o𝑓. b) Is 𝑓(𝑥)=x^3 invertible? Explain wi

Algebra ->  Functions -> SOLUTION: a) For the given functions 𝑓(𝑥)= x/x-1 and g(x)=-4/x ,find the domain of the composite functions 𝑓o𝑔 and 𝑔o𝑓. b) Is 𝑓(𝑥)=x^3 invertible? Explain wi      Log On


   

Question 1189264: a) For the given functions 𝑓(𝑥)= x/x-1 and g(x)=-4/x ,find the domain of the
composite functions 𝑓o𝑔 and 𝑔o𝑓.
b) Is 𝑓(𝑥)=x^3 invertible? Explain with reason.
If 𝑓(𝑥) is invertible, find the inverse of 𝑓(𝑥). Also, graph 𝑓 and 𝑓^-1 on the
same coordinate axes.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
a) For the given functions f%28x%29=+x%2F%28x-1%29 and g%28x%29=-4%2Fx ,find the domain of the
composite functions f o g and g o+f.
f+o g=f%28g%28x%29%29
f+o g=f%28-4%2Fx%29
f+o g=%28-4%2Fx%29%2F%28-4%2Fx-1%29
f+o g=%28-4%2Fx%29%2F%28-%28x+%2B+4%29%2Fx%29
f+o g=4%2F%28x+%2B+4%29
domain: exclude x that makes denominator equal to zero which is x=-4
so, domain is { x element R : x%3C%3E-4 }
g o f=g%28f%28x%29%29
g o f=g%28x%2F%28x-1%29%29
g o f=-4%2F%28x%2F%28x-1%29%29
g o f=-%284+%28x+-+1%29%29%2Fx
domain:{ x element R : x%3C%3E0 }

b) Is f%28x%29=x%5E3+invertible? Explain with reason.
yes, f%28x%29=x%5E3 is invertible
reason: pass a horizontal line test
If we take a horizontal line and slide it up and down the graph, it only ever intersects the function in one spot! This means that each output corresponds with exactly one input. In other words, each input has a unique output. The function f%28x%29=x%5E3 is invertible.

If f%28x%29 is invertible, find the inverse of f%28x%29. Also, graph f and f%5E-1 on the same coordinate axes.
f%28x%29=x%5E3............recall+f%28x%29=y
y=x%5E3..........swap variables
x=y%5E3.........solve for y
y=root%283%2Cx%29
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