Another Composite Functions Question.?

Question

If f(x) = 6x - 2 / 3 and g(x) is a linear function such that (g ○ f)(x) = 2x + 1, find g(x).

Answer 1

Ah well, if you meant:

f(x) = (6x - 2)/3, go with Puggy's answer.

If you really meant:

f(x) = 6x - 2/3, go with mine.

And be more careful when you post questions. :c(

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If g(x) is linear, then it can be written as:

g(x) = ax + b

Also we know that:

g(6x - 2/3) = 2x + 1

a(6x - 2/3) + b = 2x + 1

6ax + (-2/3a + b) = 2x + 1

So, equating the coefficients:

6a = 2, therefore a = 2/6 = 1/3.

-2/3a + b = 1

-2/3(1/3) + b = 1

-2/9 + b = 1

b = 11/9

So g(x) = 1/3x + 11/9.

Double-checking:

g(6x - 2/3) = 1/3(6x - 2/3) + 11/9 = 1/3(6x) - 1/3(2/3) + 11/9 = 2x - 2/9 + 11/9 = 2x + 1, check.

Answer 2

f(x) = (6x - 2)/3
g(x) is a linear function

(g o f)(x) = 2x + 1

To find g(x), we know that it is a linear function, so g(x) will be of the form

g(x) = mx + b

If we can obtain the values for m and b, we're set.
If we solve for (g o f)(x) using our defined unknown function,

(g o f)(x) = g(f(x)) =
g((6x - 2)/3) = m[ (6x - 2)/3 ] + b
= (m/3) (6x - 2) + b
= 6mx/3 - 2m/3 + b
= 2mx - 2m/3 + b
= 2mx + (-2m/3 + b)

However, we're given that (g o f)(x) = 2x + 1, which means we can equate the coefficients to solve for m and b.

The coefficient of x in our unknown function is 2m, and the coefficient of x in the given function is 2. Therefore

2m = 2, which means m = 1.

The constant is equal to 1, but the constant is also equal to
(-2m/3 + b), so

-2m/3 + b = 1. We know m = 1, so
-2(1)/3 + b = 1
-2/3 + b = 1
b = 1 + 2/3
b = 5/3

Therefore, m = 1, b = 5/3. Recall that we had this:

g(x) = mx + b. Now we know the values for m and b.

g(x) = x + 5/3

Answer 3

f(x)= 6x-(2/3) and g(x)=? and (g(f(x))= 2x+1

so by definition of composite functions... 6(g(x))-(2/3)=2x+1

6(g(x))-(2/3)=2x+1
6(g(x))= 2x+(5/3)
g(x)= (2x+(5/3))/6
or
(x/3)+(5/18)= g(x)

And to check:
Substitute g(x) into the original equation
6(x/3+5/18)-(2/3)=2x+1
(2x+5/3)-(2/3)=2x+1
2x+1=2x+1

I interpreted your problem a little differently. I hope it is correct.