1. ## Composite Functions

Will someone please solve the following composite function for g(x):

g[f(x)] = x-1; given f(x) = sqrt(x+2)

I believe the answer is (x^2) - 3, but I cannot remember how to solve using basic algebra.

TIA,
J. Lane

2. Originally Posted by Knowledge
Will someone please solve the following composite function for g(x):

g[f(x)] = x-1; given f(x) = sqrt(x+2)

I believe the answer is (x^2) - 3, but I cannot remember how to solve using basic algebra.

TIA,
J. Lane
your answer is correct. loosely speaking there is no real algebraic method to do this, it is more like a pattern recognition problem. you should just realize that to go from $\sqrt{x + 2}$ to $x - 1$ you would square and subtract 3

3. Ok then, for the following composite function f[g(x)] = [(x^4)+(x^2)]/[1+(x^2)]; given g(x) = 1 - (x^2):

How would you solve for f(x) using "pattern recognition"? I am certain there is an appropriate method to solve using algebra because I remember learning it back in the 8th grade, but as the saying goes, "if you don't use it, you lose it."

Thank you again,
J. Lane

4. Originally Posted by Knowledge
Ok then, for the following composite function f[g(x)] = [(x^4)+(x^2)]/[1+(x^2)]; given g(x) = 1 - (x^2):

How would you solve for f(x) using "pattern recognition"? I am certain there is an appropriate method to solve using algebra because I remember learning it back in the 8th grade, but as the saying goes, "if you don't use it, you lose it."

Thank you again,
J. Lane
hmm, i believe you are right. i have forgotten the method also. i will try to look it up and get back to you. the "pattern recognition" i talk about is probably using this method without realizing it. i do that sometimes. because i do something over and over, i tend to internalize it and end up doing it "naturally"--forgetting the original method or formula in the process.

here i get $f(x) = \frac {(1 - x)(2 - x)}{2 - x}$

again, recognizing patterns:

note that we have $f(g(x)) = \frac {x^2(1 + x^2)}{1 + x^2}$

for $1 - x^2 \to x^2$, take the negative and add 1, hence you get $1 - x$

for $1 - x^2 \to 1 + x^2$, take the negative and add 2, hence $2 - x$