Expert: Paul Klarreich - 7/11/2007

Question
Me again.  Now I'm on functions.    Is there a specific formula that I go by?  I think I'm supposed to get the product of the two functions, but as usual I have no idea!!  Here's the question:

For the functions f(x) = -5x + 12 and g(x) = 10x + 4,
find g(f((-1)).

Answer
Questioner:   Taneisha
Category:  Advanced Math
Question:  Me again.  Now I'm on functions.    Is there a specific formula that I go by?  I think I'm supposed to get the product of the two functions, but as usual I have no idea!!  Here's the question:

For the functions f(x) = -5x + 12 and g(x) = 10x + 4,
find g(f((-1)).  << extra parenthesis mistyped.
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Hi, Taneisha,

In working with functions, you have to be careful of your notation and interpret it correctly.  For example

1. f(x) + g(x)  is a sum of functions.  It says to add the results - the outputs - of computing f and g.

2. f(x)g(x)  is a product of functions.  It says multiply the results - the outputs - of computing f and g.

3. f(x) - g(x),  4. f(x)/g(x), etc...

BUT:

5. f(g(x)) is a composition of functions.  It says to apply g to (x); THEN use that as the INPUT of f.  
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Imagine that you have two friends, Frank and George, who we can call f and g, for short, because each is an 'expert' at computing that function.  In cases 1,2,3,4, you can take your value of x, give a copy to each of Frank and George. They work simultaneously and when you get the two answers, you add, multiply, etc. to get your answer.

But for case 5, you must FIRST give your value of x to Frank.  (In  g(f(x)), f is the inside function and is evaluated first.)  You WAIT.  When Frank gives you his answer, THEN you give THAT ANSWER to George.  Whatever he returns is your final answer.
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Your example:

f(x) = -5x + 12 and g(x) = 10x + 4,  find g(f(-1))  means:

1. First apply  f to -1:

f(-1) = -5(-1) + 12 = 5 + 12 = 17

2. Now apply g to that result:

g(17) = 10(17) + 4 = 170 + 4 = 174.

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Extra.  What if your problem said to compute g(f(x))?  Then you will say:

1. f(x) = -5x + 12

2. g(f(x)) = g( -5x + 12) =

10(-5x + 12) + 4 = -50x + 120 + 4

= -50x + 124
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Notice that

-50(-1) + 124 = 50 + 124 = 174, just as before.