Please help!! I cant find the domains

Question

For each pair of functions, find  and  . Determine the domain of  and the domain of . Show your work.



1.  and 

2.  and 

3.  and 

4.  and 

5.  and 

Answer 1

f=6x+5, g=2x-4

(fog)(x) = 6(2x-4) +5= 12x -19

domain is all reals

(Gof)(x)= 2(6x+5) -4 = 12x +6 (it's the computer that keeps turning gof into God. spooky.

domain is all reals

2nd &3rd problems have some limits on the domain due to the denominator otherwise being zero and the square root of a negative number

Answer 2

In case you don't know, f_°g(x) means f(g(x)) or, putting the function "g" into the function "f" in place of the "x".

And g_°f(x) means g(f(x)).

1) f(g(x)) means 6(2x-4) + 5 and there are no restrictions for "x" so the domain is all real numbers or (-∞,∞) and g(f(x)) means 2(6x+5) - 4 and likewise, there are no limitations as to what values you can use for "x" therefore, again, he domain is all real numbers or (-∞,∞).

2) f(g(x)) means 4(√(x+7))² - 16. Square roots are restricted (for real numbers) to be greater than or equal to zero therefore x + 7 > 0 which means that the domain must be greater than or equal to -7 or [-7,∞)

g(f(x)) means √((4x²-16)+7) and for the same reasons, (4x²-16)+7 > 0 which requires a little algebra:

(4x²-16)+7 > 0

4x² - 9 > 0

4x² > 9

x² > 9/4

The critical values here are when x = 3/2 and when x = -3/2. Making a number line divides this into 3 regions, those less than -3/2, those between -3/2 and 3/2 and those greater than 3/2. Trying values in each of those regions shows that the region between -3/2 and 3/2 does not make the inequality true therefore this region is restricted from the domain. Therefore the domain is (-∞,-3/2) U (3/2,∞)

3) f(g(x)) means 5/[(4x+4)-4]. Rational functions have the restriction that the denominator cannot be zero. Therefore (4x+4)-4 ≠ 0 or 4x≠0 or x≠0 therefore the domain is all real numbers except zero or (-∞,0)U(0,∞).

g(f(x)) means 4[5/(x-4)] + 4 and for the same reasons x-4≠0 or x≠4 therefore the domain is all real numbers except 4 or (-∞,4)U(4,∞)

Continue on in the same way looking for any restrictions in "x" after you put the functions together. In problems 4 and 5 you will not find any restrictions.