Composition of Functions: f(x) = 2x^2+1 g(x) = 1-2x --> Solve for g(f(1))

[dnymeyer]

this is a simple question and I feel like I should be able to figure it out myself, but I'm finding conflicting answers, so please forgive me for the simplicity of this question...

I understand composition of functions such as f(g(x)) is basically plugging g(x) into f(x) equation.
What I don't understand, is the order for solving...

For example:
f(x) = 2x^2+1
g(x) = 1-2x
--> Solve for g(f(1))

If you solve f(1) equation and then plug the answer into g(x), you get
f(1) = 2x^2 + 1
Solves to 3....

plug into g(x) = 1-2(3)
Solves to 5.

But if simply plug f(1) into g(x) and then solve, you get:
1-2(2(1)^2 +1)
Solves to -3

Which is the correct order?

 

[dnymeyer]

Oh my gosh! How embarrassing! After looking over this question again and again, I didn't realize I did my math wrong until I actually made this post..

How embarrassing! Sorry, I would delete this thread if I knew how. :-x

But since I've already posted, just to clarify, you can solve equations like this in any order?

 

[JeffM]

this is a simple question and I feel like I should be able to figure it out myself, but I'm finding conflicting answers, so please forgive me for the simplicity of this question...

I understand composition of functions such as f(g(x)) is basically plugging g(x) into f(x) equation.
What I don't understand, is the order for solving...

For example:
f(x) = 2x^2+1
g(x) = 1-2x
--> Solve for g(f(1))

If you solve f(1) equation and then plug the answer into g(x), you get
f(1) = 2x^2 + 1
Solves to 3....

plug into g(x) = 1-2(3)
Solves to 5.

But if simply plug f(1) into g(x) and then solve, you get:
1-2(2(1)^2 +1)
Solves to -3

Which is the correct order?

The notation helps you if you keep in mind how it is to be interpreted.

f(x) tells you to apply the rule designated by f to x, whatever x is. So f(g(x)) means to apply the f rule to the result of applying the g rule to x.

g(x) tells you to apply the rule designated by g to x, whatever x is. So g(f(x)) tells you to apply the g rule to the result of appkying the f rule to x.

\(\displaystyle \text {Let } y = f(x) \text { and } z = g(x) \implies\)

\(\displaystyle f(g(x)) = f(z) \text { and } g(f(x)) = g(y).\)

Continued in next post.

 

[JeffM]

this is a simple question and I feel like I should be able to figure it out myself, but I'm finding conflicting answers, so please forgive me for the simplicity of this question...

I understand composition of functions such as f(g(x)) is basically plugging g(x) into f(x) equation.
What I don't understand, is the order for solving...

For example:
f(x) = 2x^2+1
g(x) = 1-2x
--> Solve for g(f(1))

If you solve f(1) equation and then plug the answer into g(x), you get
f(1) = 2x^2 + 1
Solves to 3....

plug into g(x) = 1-2(3)
Solves to 5.

But if simply plug f(1) into g(x) and then solve, you get:
1-2(2(1)^2 +1)
Solves to -3

Which is the correct order?

I find it more useful to think of a function as a rule of operations rather than an equation (Of course, it is an equation in form, but its importance lies in telling you what to do to the function's argument.)

\(\displaystyle \text {Given: }f(x) = 2x^2 + 1 \text { and } g(x) = 1 - 2x \implies\)

\(\displaystyle f(g(x)) = f(1 - 2x) = 2(1 - 2x)^2 + 1 = 2(1 - 4x + 4x^2) + 1 = 3 - 8x + 8x^2, \text { and}\)

\(\displaystyle g(f(x)) = g(2x^2 + 1) = 1 - 2(2x^2 + 1) = -\ (4x^2 + 1).\)

\(\displaystyle f(g(1) = f(-\ 1) = 3 = 3 - 8(1) + 8(1^2).\)

\(\displaystyle g(f(1)) = g(3) = -\ 5 = -\ ( 4 * 1^2 + 1).\)

To the extent these two posts leave you with further questions, please do not hesitate to ask.

 

[Dr.Peterson]

I understand composition of functions such as f(g(x)) is basically plugging g(x) into f(x) equation.
What I don't understand, is the order for solving...

For example:
f(x) = 2x^2+1
g(x) = 1-2x
--> Solve for g(f(1))

If you solve f(1) equation and then plug the answer into g(x), you get
f(1) = 2x^2 + 1
Solves to 3....

plug into g(x) = 1-2(3)
Solves to -5.

But if simply plug f(1) into g(x) and then solve, you get:
1-2(2(1)^2 +1)
Solves to -5

Which is the correct order?

Presumably you already made the two fixes I made above, so you know that both methods are fine.

But we need to deal with your terminology, which can get in the way of understanding. You are not "solving the f(1) equation"; you are evaluating the expression for f(1). An equation contains an equal sign, and solving means finding the value of the variable -- it's the opposite of what is happening here.

The only difference in the two things you did (when done right) is in whether you evaluated 2x^2 + 1 before or after doing the substitution -- that is, whether you substituted that for f(1) before or after finding that it is 3. Each approach can be easiest in different situations; and when you don't have a specific value for x, you have to do it the second way and simplify rather than evaluate, so you need to learn to do that well anyway.

I sometimes do both ways to check my work.