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Solve for domain. Solve for x in log and ln function

1. Given that f(x) = sqrt(3x+6) and g(x) = x-4/x, find the following functions and express their domains in interval notation.

a) (f+g)(x)=________, its domain is _______
b) (f/g)(x)=_______, its domain is _______
c) (g/f)(x)=_______, its domain is _______
d) (f ○ g)(x)=_______, its domain is_______

2. Solve the following equations.

a) log'4 (6x-4)=2
x=_____
b) log'4 6/x+7=-2
x=_____
c)5^2x+3 =21
x=_____

3.Consider the function y=Aa^x, where A and a are positive constants.
Note. Aa^x means A(a^x) and not (Aa)^x

a) For this part of the question, suppose that whenever the value of x is increased by 2, the value of y is doubled. Determine the values of A and a. Enter "Indeterminate" if the value(s) cannot be uniquely determined.
A=_____, and a=_____

b) For this part of the question, suppose that the y− intercept of the graph of this function is 8. Determine the values of A and a. Enter "Indeterminate" if the value(s) cannot be uniquely determined.
A=_____, and a=_____

(c) Suppose now that both of the conditions described in parts (a) and (b) are satisfied. Find the value of x for which y is equal to 64.
x=______

4. Consider the function f(x)=19 In(4x-7). Answer the following

a) The domain of f is the set ________.

b)The x- intercept of the graph of f is ______. If the graph does not have an x- intercept, enter DNE.

c) Find the following values. Enter DNE if the answer is undefined or does not exist.
f(22/5)=_____
f(2/9)=_____
f7/4)=_____
The value of x for which f(x)=15 is ______. If no such value exists for x, enter DNE

d) The equation of the vertical asymptote of the graph of f is x=______> if the graph does not have a verticl asymptote, enter DNE.

Solution Preview

Please refer to the attached file. Cheers.

Simple introductory calculus questions
1. Given that f(x) = sqrt(3x+6) and g(x) = x-4/x, find the following functions and express their domains in interval notation.

a) (f+g)(x)=________, its domain is _______
b) (f/g)(x)=_______, its domain is _______
c) (g/f)(x)=_______, its domain is _______
d) (f ○ g)(x)=_______, its domain is_______

f(x)=√(3x+6)
The domain of f(x) is
3x+6≥0
3x≥-6
x≥-2

g(x)=x-4/x
The domain of g(x) is
x≠0
(a)
(f+g)(x)=f(x)+g(x)
=√(3x+6)+x-4/x
The domain is -2  x < 0 or x > 0.
[-2, 0)  (0, )

(b)
(f/g)(x)=√(3x+6)/(x-4/x)

x-4/x≠0
x≠4/x
x^2≠4
x≠±2

The domain is -2 < x < 0, 0 < x < 2, x > 2
(-2, 0)  (0, 2)  (2, )

(c)
(g/f)(x)=(x-4/x)/√(3x+6)

3x+6≠0
x≠-2

The domain is -2 < x < 0 or x > 0.
(-2, 0)  (0, )

(d)
(fog)(x)=√(3(x-4/x)+6)
=√(3x-14/x+6)

3x-14/x+6≥0
For x > 0,
3x^2-14+6x≥0
3x^2+6x-14≥0

For 3x2 + 6x - 14 = 0, the solution is x= 1.38 or -3.38. The ...

Solution Summary

In question 1, students are asked to find the domain of (f+g)(x), (f/g)(x), (g/f)(x), and fog(x). In question 2, several logarithm functions are involved. In question 3, more complicated problem involving the leading coefficient and the base of an exponential function is solved. In question 4, a ln function is explored in detail.

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