How to find g(x) if f(x) and gof(x) is given

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This becomes way easier if you have the knowledge of inverse functions.

Let me make this clear. You have $f(x) = x^2 + 1$ and $g(f(x)) = 1/(x^2 + 4)$. Now pause and think about the second function. The function is defined as $g(f(x))$, right. now what if there is some way that you could manipulate this function and some how change it to $g(x)$. Or think about with what other function should you multiply $f(x)$ [the function in $g(f(x))$ ] to get $x$ [so that the function would be $g(x)$]. It is $f(x)^{-1}$ right.

So let us find $f(x)$ inverse. $f(x) = x^2 +1$ then $f(x)^{-1} = sqrt{x – 1}$ right [ just swipe the $x$ with $f(x)$ and solve for $f(x)$ that should give you this result]now go to the first function $g(f(x))$ [this function is the same as $g(x^2 + 1)$ right.]

Now find $g(sqrt{x -1})$ (that is $g$ of the inverse function you have found) in the function which is $g( x^2 + 1)$.

Thus the result becomes$$g((sqrt{x – 1})^2 + 1) = frac{1}{(sqrt{x – 1})^2 + 4}.$$Now here comes the beauty: what is $(sqrt{x-1})^2 + 1$? It is $x$, so the value of $g((sqrt{x-1})^2 + 1) = g(x)$.Now $$g(x) = frac{1}{(sqrt{x – 1})^2 + 4} = frac{1}{x + 3 }.$$

Method and examples

Composite functions and Evaluating functions : f(x), g(x), fog(x), gof(x) calculator

 Method  1. Find Range of `f:A->B`f(x) =A =

A

=,x `in` , x is

  1. `f(x)=5x+2` and `A={1<=x<5; x in N; x is odd}`
  2. `f(x)=3x-2` and `A={x in N}`
  3. `f(x)=|2x+1|` and `A={x in Z}`
  4. `f(x)=x^3` and `A={-2<=x<=2}`
  5. `f(x)=sqrt(x)` and `A={1,4,16,36}`
  6. `f(x)=x^2` and `A={[1,5]}`
  7. `f(x)=(-2)^x` and `A={x in N}`
  8. `f(x)=sin(x)` and `A={0,30,90}`
  9. `f(x)=(x^2+1)/(x+1)` and `A={-2,0,2}`
  10. `f(x)=x^2-2x+3` and `A={-1,3}`

2. Composite functions and Evaluating functions : f(2), g(3), fog(x), gof(x), fof(x), (f+g)(x)==Find()Also Find  

  1. `f(x)=x-1, g(x)=3x+5`. Find fog(x).
  2. `f(x)=x+3, g(x)=x^2`. Find gof(x).
  3. `f(x)=3x+1, g(x)=-x^2+5`. Find gof(x).
  4. `f(x)=4x+1, g(x)=2x-3`. Find gof(2).
  5. `fog(x)=(x+2)/(3x), f(x)=x-2`. Find g(2).
  6. `gof(x)=(x+2)/(3x), g(x)=x-2`. Find f(x).
  7. `fog(x)=2x^2-1, f(x)=2+x^2`. Find g(x).
  8. `gof(x)=1/x^2, f(x)=2+x^2`. Find g(x).

3. Find valuef(x) =g(x) =h(x) =Find 

  1. `f(x)=x(x+1)(2x+1)`. Find f(x)-f(x-1).
  2. `f(x)=1/(x+1)`. Find f(-x)-f(x).
  3. `f(x)=1/x`. Find f(x+1)-f(x-1).
  4. `f(x)=x^2-x`. Find f(x+1)-f(x).
  5. `f(x)=2x-3`. Find f(0)+f(1)+f(2).
  6. `f(x)=x^2-2^x`. Find f(2)-f(0).
  7. `f(x)=(x^2+1)/(x^3-x+1)`. Find f(1)-f(0).
  8. `f(x)=2x-6`. Find f(a-3)+f(a).
  9. `f(x)=(x-2)^2`. Find f(x+2).

4. Verifying if two functions are inverses of each otherf(x) =g(x) = 

  1. `f(x)=x+3`. `g(x)=x-3`.
  2. `f(x)=4x-3`. `g(x)=(x+3)/4`.
  3. `f(x)=x/(x-1)`. `g(x)=(2x)/(2x-1)`.

Domain, Range, Inverse, Vertex, Symmetry, Directrix, Intercept, Parity, Asymptotes of a function calculator

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Written by Jane