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Let # f(x)= -4x+3# and #g(x)= 1/(x+2)#, how do you evaluate f(g(x)) and g(f(x)) for x?

Answer 1

Elias Ackerman

Redefine the equations as follows:

#g(f(x))=1/(f(x)+2)=1/(5-4x)#

#f(g(x))=-4g(x)+3=3-4/(x+2)#

You can then evaluate the new functions using these definitions.

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Answer 2

Xavier Alonso

To evaluate ( f(g(x)) ) and ( g(f(x)) ) for ( x ), follow these steps:

Substitute the function ( g(x) ) into the function ( f(x) ) to evaluate ( f(g(x)) ).
Substitute the function ( f(x) ) into the function ( g(x) ) to evaluate ( g(f(x)) ).

Let's compute these:

Evaluate ( f(g(x)) ): [ f(g(x)) = f\left(\frac{1}{x+2}\right) = -4\left(\frac{1}{x+2}\right) + 3 = -\frac{4}{x+2} + 3 ]
Evaluate ( g(f(x)) ): [ g(f(x)) = g(-4x + 3) = \frac{1}{(-4x + 3) + 2} = \frac{1}{-4x + 5} ]

So, ( f(g(x)) = -\frac{4}{x+2} + 3 ) and ( g(f(x)) = \frac{1}{-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.