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If #f(x)=x^2+4x# and #g(x)=3x-5#, how do you find #(f(g(x))# and #g(f(x))#?

Answer 1

It might be better if we think of the functions as having different parameters, so

If #f(a)=a^2+4a# and #g(b)=3b−5#,

then #f(g(x))# simply means re-writing the #f(a)# equation with #a# replaced by #g(x)#: #f(g(x))# # = (g(x)^2 + 4(g(x))# # = (3x - 5)^2 + 4(3x -5)# # = 9x^2 -18x +5#

Similarly #g(fx))# # = 3(x^2 + 4x) - 5# (by replacing #b# with #f(x)#) # = 3x^2 + 12x#

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Answer 2

Theodore Amidon

To find (f(g(x))), you substitute the expression for (g(x)) into the function (f(x)), and then simplify.

[f(g(x)) = f(3x - 5) = (3x - 5)^2 + 4(3x - 5)]

To find (g(f(x))), you substitute the expression for (f(x)) into the function (g(x)), and then simplify.

[g(f(x)) = g(x^2 + 4x) = 3(x^2 + 4x) - 5]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.