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