Question 1168502
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If you use the "√" symbol in your post, use parentheses to make the meaning clear.<br>
Is H(x) equal to √6x-1 = {{{sqrt(6x)-1}}}?
Or is it equal to √(6x-1) = {{{sqrt(6x-1)}}}?<br>
I suspect it is supposed to be the second; but it could be either one.<br>
I will go ahead and explain how to answer the question if {{{H(x) = sqrt(6x)-1}}}.<br>
If it is the other one, you can use my discussion below to answer the question for that other case.<br>
Look at the function and see what it does to the input value x:
(1) multiply by 6;
(2) take the square root; and
(3) subtract 1<br>
To make the function H(x) a composition of two function f(x) and g(x), simply condense that sequence of three operations into a sequence of two operations, keeping all the operations in the original order.<br>
There are two possibilities:<br>
One choice is to combine steps (1) and (2):
(1) multiply by 6 and take the square root; and
(2) subtract 1<br>
Then the function g(x) is the first of those steps: {{{g(x) = sqrt(6x)}}};
and the function f(x) is the second: {{{f(x) = x-1}}}<br>
ANSWER (one possibility): {{{f(x) = x-1}}} and {{{g(x) = sqrt(6x)}}}<br>
The other choice is to keep step 1 by itself and combine steps 2 and 3:
(1) multiply by 6; and
(2) take the square root and subtract 1<br>
Then the function g(x) is {{{g(x) = 6x}}};
and the function f(x) is {{{f(x) = sqrt(x)-1}}}<br>
ANSWER (a second possibility): {{{f(x) = sqrt(x)-1}}} and {{{g(x) = 6x}}}<br>