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How do you find the compositions given # f(x)=8x-1# and #g(x)=x/2#?

Answer 1

To find the composition of two functions, denoted as ( f(g(x)) ) or ( g(f(x)) ), you substitute the expression for one function into the other function.

For ( f(g(x)) ): [ f(g(x)) = f\left(\frac{x}{2}\right) ] [ = 8\left(\frac{x}{2}\right) - 1 ] [ = 4x - 1 ]

For ( g(f(x)) ): [ g(f(x)) = g(8x - 1) ] [ = \frac{8x - 1}{2} ] [ = 4x - \frac{1}{2} ]

So, the compositions are: [ f(g(x)) = 4x - 1 ] [ g(f(x)) = 4x - \frac{1}{2} ]

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Answer 2

Lily Anderson

#g(f(x))=g@f=(8x-1)/2=4x-1/2#
#f(g(x))=f@g=8x/2-1=4x-1#

You can think: a) #y=f(x)=8x-1# #z=g(y)=y/2# #z=h(x)=g(f(x))=(8x-1)/2=4x-1/2#

b) #y=g(x)=x/2# #z=f(y)=8y-1# #z=h(x)=f(g(x))=8x/2-1=4x-1#

Remember that:

#g@f!=f@g#

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Answer 3

Chloe Alexander

Substitute the expression for #g(x)# in place of #x# in the definition of #f(x)# to find: #(f@g)(x) = 4x-1#

Similarly find: #(g@f)(x) = 4x-1/2#

#(f@g)(x) = f(g(x)) = f(x/2) = 8(x/2)-1 = 4x-1#

#(g@f)(x) = g(f(x)) = g(8x-1) = ((8x-1))/2 = 4x-1/2#

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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.