Calculus Question?

Question

Is the function f(x)=x+2 identical to the function
g(x)=((x^2)+5x+6)/(x+3)

Why or why not?

And what is the inverse of f(x)=(2x^3)+5

Answer 1

no. g(x) is discontinuous at x=-3
f^-1(x)=((x-5)/2)^1/3

Answer 2

OK. The first part:
g(x) = (x^2+5x+6)/(x+3).
Factorise the quadratic on top:
x^2+5x+6 = (x + 3) (x + 2).
Hence, (x+3) can cancel from both the numerator and the denominator:
g(x) = (x+3)(x+2)/(x+3) becomes g(x) = (x+2).
HOWEVER I am not sure. You may want to check the definition of 'identical' with your teacher. While the functions may appear to be the same, g(x) is discontinuous at x = -3 but f(x) is not.

f(x) = 2x^3 + 5.
To find the inverse of a function, you want to get f(x) in terms of x. To make it simpler, I will use a substitution, y = f(x).

y = 2x^3 + 5, aim: to make x the subject.
2x^3 = y - 5
x^3 = (y - 5) / 2
x = [(y -5)/2]^(1/3)
(or x = cube root of the whole fraction (y-5)/2)

Hence, the inverse function f^-1(x) is [(x-5)/2]^1/3.

Answer 3

The graphs of the two equations are identical, except there is a hole at x=-3.

y = f(x)=(2x^3)+5
x =[(y-5)/2}^1/3 <-- inverse

Answer 4

g(x)=(x+3)(x+2)/((x+3) which is defined for all x except -3. Which means you can put in any x except -3 into g(x) and come up with a value.
f(x) = x + 2 is defined for all values of x including x = -3. So, that makes them different at x= -3.

Answer 5

Yes. g(x) = (x^2+5x+6)/(x+3) = (x+2)(x+3)/(x+3) = x+2

inverse funciton --> x = sqrt((f(x)-5)/2)

Answer 6

g(x) = ((x+3)(x+2))/(x+3)

g(x) = (x+2)

they are identical.