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
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 $\displaystyle f(x) = \frac {(1 - x)(2 - x)}{2 - x}$
again, recognizing patterns:
note that we have $\displaystyle f(g(x)) = \frac {x^2(1 + x^2)}{1 + x^2}$
for $\displaystyle 1 - x^2 \to x^2$, take the negative and add 1, hence you get $\displaystyle 1 - x$
for $\displaystyle 1 - x^2 \to 1 + x^2$, take the negative and add 2, hence $\displaystyle 2 - x$