How do you find #(f @ g)(x)# and its domain, #(g @ f)(x)# and its domain,# (f @ g)(-2) # and #(g @ f)(-2)# of the following problem #f(x) = x+ 2#, #g(x) = 2x^2#?

Answer 1

To find ( (f @ g)(x) ), ( (g @ f)(x) ), ( (f @ g)(-2) ), and ( (g @ f)(-2) ) for the functions ( f(x) = x + 2 ) and ( g(x) = 2x^2 ), follow these steps:

  1. ( (f @ g)(x) ) is obtained by substituting ( g(x) ) into ( f(x) ). So, ( (f @ g)(x) = f(g(x)) = f(2x^2) = 2x^2 + 2 ).

    The domain of ( (f @ g)(x) ) is the set of all real numbers since there are no restrictions on the domain of ( 2x^2 + 2 ).

  2. ( (g @ f)(x) ) is obtained by substituting ( f(x) ) into ( g(x) ). So, ( (g @ f)(x) = g(f(x)) = g(x + 2) = 2(x + 2)^2 = 2(x^2 + 4x + 4) = 2x^2 + 8x + 8 ).

    The domain of ( (g @ f)(x) ) is also the set of all real numbers.

  3. To find ( (f @ g)(-2) ), substitute ( -2 ) into ( 2x^2 + 2 ): ( (f @ g)(-2) = 2(-2)^2 + 2 = 2(4) + 2 = 10 ).

  4. To find ( (g @ f)(-2) ), substitute ( -2 ) into ( 2x^2 + 8x + 8 ): ( (g @ f)(-2) = 2(-2)^2 + 8(-2) + 8 = 2(4) - 16 + 8 = 16 - 16 + 8 = 8 ).

So, ( (f @ g)(-2) = 10 ) and ( (g @ f)(-2) = 8 ).

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Answer 2

See the explanation below...

By the composition #(f\circ g)(x)#, we mean #f(g(x))# and by #(g\circ f)(x)#, we mean #g(f(x))#.
To find #f(g(x))#, we need to put the value of #g(x)# for every value of #x# in #f(x)#. So, by doing this, we get:
#(f\circ g)(x)=f(g(x))=(2x^2)+2#
#=2x^2+2#

A function's domain is the range of values that the function is defined and real for.

This above evaluated function has no undefined points. The domain is # -oo < x < oo #

Likewise, we have:

#(g\circ f)(x)=g(f(x))=2(x+2)^2#

Simplify:

#=2x^2+8x+8#
Domain: # -oo < x < oo #
Now, in the same manner, find #(f\circ g)(-2)# as:
#=2(-2)^2+2#

Simplify:

#=10#

And:

#(g\circ f)(-2)=2(-2)^2+8(-2)+8#

Simplify:

#=0#
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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.

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.

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.

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.

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