Question

Let f(x) = 7 - 2x and g(x)= x+3.

(a) Find (g*f)(x).

(b) Write down g^-1(x).

(c) Find (f*g^-1)(5).

[this symbol * means multiplication and this symbol ^ means power]

Answer 1

Answer:

please mark it as brianlist

Step-by-step explanation:

Given f(x) = 2x + 3 and g(x) = –x2 + 5, find ( f o g)(x).

In this case, I am not trying to find a certain numerical value. Instead, I am trying to find the formula that results from plugging the formula for g(x) into the formula for f(x). I will write the formulas at each step, using parentheses to indicate where the inputs should go:

( f o g)(x) = f (g(x))

   = f (–x2 + 5)

   = 2(             ) + 3     ... setting up to insert the input formula

   = 2(–x2 + 5) + 3

   = –2x2 + 10 + 3

   = –2x2 + 13

If you plug in "1" for the x in the above, you will get ( f o g)(1) = –2(1)2 + 13 = –2 + 13 = 11, which is the same answer we got before. Previously, we'd plugged a number into g(x), found a new value, plugged that value into f(x), and simplified the result. This time, we plugged a formula into f(x), simplified the formula, plugged the same number in as before, and simplified the result. The final numerical answers were the same. If you've done the symbolic composition (the composition with the formulas) correctly, you'll get the same values either way, regardless of the value you pick for x. This can be a handy way of checking your work.

Here's another symbolic example:   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

Given f(x) = 2x + 3 and g(x) = –x2 + 5, find (g o f )(x).

(g o f )(x) = g( f(x))

   = g(2x + 3)

   = –(           )2 + 5    ... setting up to insert the input

   = –(2x + 3)2 + 5

   = –(4x2 + 12x + 9) + 5

   = –4x2 – 12x – 9 + 5

   = –4x2 – 12x – 4

There is something you should note from these two symbolic examples. Look at the results I got:

( f o g)(x) = –2x2 + 13

(g o f )(x) = –4x2 – 12x – 4

That is, ( f o g)(x) is not the same as (g o f )(x). This is true in general; you should assume that the compositions ( f o g)(x) and (g o f )(x) are going to be different. In particular, composition is not the same thing as multiplication. The open dot "o" is not the same as a multiplication dot "•", nor does it mean the same thing. While the following is true:

f(x) • g(x) = g(x) • f(x)            [always true for multiplication]

...you cannot say that:

( f o g)(x) = (g o f )(x)           [generally false for composition]

That is, you cannot reverse the order in composition and expect to end up with the correct result. Composition is not flexible like multiplication, and is an entirely different process. Do not try to multiply functions when you are supposed to be plugging them into each other.

Answer 2

Answer:

Given $f(x)=7-2x , g(x)=x+3$,

$(g$×$f)$$(x)=-2x^{2} +x+21$

$g^{-1}(x)=x-3$

$(f$×$(g^{-1} ))(5)=-6$

Step-by-step explanation:

Given  $f(x)=7-2x , g(x)=x+3$

(a)

$(f$×$g)(x)= f(x)g(x)$

therefore  $(g$×$f)(x)=g(x)f(x)=(x+3)(7-2x)$

                                                       $= 7x+21-2x^{2} -6x$

                                                        $=-2x^{2} +x+21$.

(b)

given $g(x)=x+3$    therefore      $g^{-1} (x)$  is

$g^{-1} (x)=x-3$.

(c)

$(f(g^{-1}))(5)=f(5)(g^{-1})(5)\\ =(7-2(5))((5-3))\\ =(7-10)(2)\\ =-6$.

thank you