How do I solve this?
- Thread starter khris le
- Start date
This is a very simple idea with a notation that takes getting used to. Let’s try this problem that chains functions together.
[math]\text {If } f(y) = y^2 + 1 \text { and } y = g(x) = 2x - 3 \\ \text {then } f(y) = f(g(x)) = \text {WHAT?}[/math]
[math]\text {If } f(y) = y^2 + 1 \text { and } y = g(x) = 2x - 3 \\ \text {then } f(y) = f(g(x)) = \text {WHAT?}[/math]
Otis
Elite Member
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- Apr 22, 2015
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Hi khris. If you're asking about the product of functions f and g, then multiply them together. That will give (fg)(x).
If you're asking about a composition of function f and g (read outloud as, "f of g of x"), then we type that as (fog)(x).
[imath]\;[/imath]
It is also shown as f(g(x)).Hi khris. If you're asking about the product of functions f and g, then multiply them together. That will give (fg)(x).
If you're asking about a composition of function f and g (read outloud as, "f of g of x"), then we type that as (fog)(x).
[imath]\;[/imath]
Thank you. I'm talking about composition of functions. I have a final math exam soon and I really need to understand this completely if I want to pass.Hi khris. If you're asking about the product of functions f and g, then multiply them together. That will give (fg)(x).
If you're asking about a composition of function f and g (read outloud as, "f of g of x"), then we type that as (fog)(x).
[imath]\;[/imath]
is it f(y)=2x-3^2
I think i've confused myself even more because this just looks horribly wrong. I'll try to think it through again and try it. Thanks for your input btw.
Otis
Elite Member
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- Apr 22, 2015
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Hi khris. Your work shows a few issues.
f(y) = y^2 + 1
Symbol y is the input variable, and f is the function name. The symbol f(y) represents the output of function f when y is the input. The right-hand side above shows what function f does with its inputs to produce corresponding outputs. Think of it like this:
f(input) = input^2 + 1
In Jeff's example, there is another function called g, and it's a function of x. The definition of function g's output is:
g(x) = 2x - 3
Now, the function composition f(g(x)) means that we're using the output of function g as the input to function f.
In other words, x goes into function g, and the value 2x-3 comes out. That output then goes into function f, and the value f(2x-3) or f(g(x)) comes out.
To write the definition for f(g(x)) when g(x) is 2x-3, we again think about what function f does with its inputs:
f(input) = input^2 + 1
The input is 2x-3, so we see that 2x-3 gets squared and then 1 is added to the result.
The first line of your work correctly shows (2x-3)^2 on the right, but don't write f(y) on the left now. You're working with a composition of two functions, so write f(2x-3) or f(g(x) on the left instead of f(y). Also, what else does function f do besides squaring its inputs? (You forgot to write that part.)
Next, when multiplying out (2x-3)^2, you haven't actually squared the expression 2x-3. What you've written out instead is ((2x)(-3))^2.
Here is the pattern for squaring a binomial expression:
(a + b)^2 = a^2 + 2ab + b^2
Or, we can view it as double distribution and get the same result:
(a + b)^2 =
(a + b)(a + b) =
a(a + b) + b(a + b) =
a^2 + ab + ab + b^2 =
a^2 + 2ab + b^2
Lastly, you had changed output symbol f(y) to input symbol y at the end, followed by solving for x in terms of y. Those steps are not part of this exercise, and we may not change function notation f(input) to input. Those two symbols are never the same.
The answer for "f of g of x" will take the form:
f(g(x)) = an expression containing x
Please try again, and let us know if you have any questions.
[imath]\;[/imath]
f(y) = y^2 + 1
Symbol y is the input variable, and f is the function name. The symbol f(y) represents the output of function f when y is the input. The right-hand side above shows what function f does with its inputs to produce corresponding outputs. Think of it like this:
f(input) = input^2 + 1
In Jeff's example, there is another function called g, and it's a function of x. The definition of function g's output is:
g(x) = 2x - 3
Now, the function composition f(g(x)) means that we're using the output of function g as the input to function f.
In other words, x goes into function g, and the value 2x-3 comes out. That output then goes into function f, and the value f(2x-3) or f(g(x)) comes out.
To write the definition for f(g(x)) when g(x) is 2x-3, we again think about what function f does with its inputs:
f(input) = input^2 + 1
The input is 2x-3, so we see that 2x-3 gets squared and then 1 is added to the result.
The first line of your work correctly shows (2x-3)^2 on the right, but don't write f(y) on the left now. You're working with a composition of two functions, so write f(2x-3) or f(g(x) on the left instead of f(y). Also, what else does function f do besides squaring its inputs? (You forgot to write that part.)
Next, when multiplying out (2x-3)^2, you haven't actually squared the expression 2x-3. What you've written out instead is ((2x)(-3))^2.
Here is the pattern for squaring a binomial expression:
(a + b)^2 = a^2 + 2ab + b^2
Or, we can view it as double distribution and get the same result:
(a + b)^2 =
(a + b)(a + b) =
a(a + b) + b(a + b) =
a^2 + ab + ab + b^2 =
a^2 + 2ab + b^2
Lastly, you had changed output symbol f(y) to input symbol y at the end, followed by solving for x in terms of y. Those steps are not part of this exercise, and we may not change function notation f(input) to input. Those two symbols are never the same.
The answer for "f of g of x" will take the form:
f(g(x)) = an expression containing x
Please try again, and let us know if you have any questions.
[imath]\;[/imath]
Steven G
Elite Member
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- Dec 30, 2014
- Messages
- 14,383
((2x)*(-3))2 = 2x*2x*(-3)(-3) = 4x^2*9 = 36x^2. You do not add the 2x with the 2x, you multiply the 2x by the 2x. Just like 7*9 is not 16 = 7+9, 2x*2x does not equal 2x+2x.
You need to learn that (a-b)2 = a2-2ab+b2