fucntion of

[spacewater]

problem
determine algebraically whether \(\displaystyle (f \circ g)(x) = (g\circ f)(x)\)
\(\displaystyle f(x) = |x|, g(x) = 2x^3\)

answer
\(\displaystyle f(g(x)) = 2x^3\)
\(\displaystyle g(f(x)) = 2x^3\)

So they are equal from my point of view but the answer sheet is showing different polynomial... Can someone point out what I missed?

 

[pka]

You must be careful with notation.
Which is it, function multiplication or function composition?
The first below is function multiplication; the second is function composition.
The proposed answer suggests function composition.
\(\displaystyle \left( {h \cdot g} \right)(x) = h(x) \cdot g(x)\;\& \,\left( {h \circ g} \right)(x) = h\left( {g(x)} \right)\).
Which is it?

 

[spacewater]

You must be careful with notation.
Which is it, function multiplication or function composition?
The first below is function multiplication; the second is function composition.
The proposed answer suggests function composition.
\(\displaystyle \left( {h \cdot g} \right)(x) = h(x) \cdot g(x)\;\& \,\left( {h \circ g} \right)(x) = h\left( {g(x)} \right)\).
Which is it?

sorry for the confusion it is



determine algebraically whether \(\displaystyle (f \circ g)(x) = (g\circ f)(x)\)
\(\displaystyle f(x) = |x|, g(x) = 2x^3\)

 

[pka]

determine algebraically whether \(\displaystyle (f \circ g)(x) = (g\circ f)(x)\)
\(\displaystyle f(x) = |x|, g(x) = 2x^3\)

Well then:
\(\displaystyle \left( {f \circ g} \right)(x) = \left| {2x^3 } \right| = 2\left| x \right|^3 \;\& \,\left( {g \circ f} \right)(x) = 2\left| x \right|^3 \,\,\left( {\left| {x^n } \right| = \left| x \right|^n } \right)\)
Note you must use absolute value.

 

[spacewater]

determine algebraically whether \(\displaystyle (f \circ g)(x) = (g\circ f)(x)\)
\(\displaystyle f(x) = |x|, g(x) = 2x^3\)

Well then:
\(\displaystyle \left( {f \circ g} \right)(x) = \left| {2x^3 } \right| = 2\left| x \right|^3 \;\& \,\left( {g \circ f} \right)(x) = 2\left| x \right|^3 \,\,\left( {\left| {x^n } \right| = \left| x \right|^n } \right)\)
Note you must use absolute value.

My answer sheet is telling me the answer is \(\displaystyle (f \circ g)(x) =|2x + 2| ; (g \circ f)(x) = 2|x+3|-1\)
I dont understand where +2 and +3|-1 came from

 

[pka]

My answer sheet is telling me the answer is \(\displaystyle (f \circ g)(x) =|2x + 2| ; (g \circ f)(x) = 2|x+3|-1\) I dont understand where +2 and +3|-1 came from

Neither do I.
Those are impossible answers if you have give us the correct functions.