MATH-Written Assignment Unit 4
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Northern Virginia Community College *
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Jan 9, 2024
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Written Assignment Unit 4
What can be said about the domain of the function
where
and
? Express it in terms of a union of intervals of real numbers. Go to
www.desmos.com/calculator
and obtain the graph of
,
, and
.
For f °g, solve f(g(x)). The f(y) will be changed to f(x), so that it would be possible to solve the composite function.
f(x) = 4 / (x – 2)
g(x) = 5 / (3x – 1)
f(g(x)) = 4 / [(5 / (3x – 1)) – 2]
= 4 / [(5 / 3x – 1) – 2(3x – 1) / (3x – 1)] = 4 / [(-6x + 7) / (3x – 1)] = 4(3x – 1) / (-6x + 7)
f(g(x)) = (12x – 4) / (-6x + 7)
“The domain of the composite function f °g is all x such that x is in the domain of g and g(x) is in the domain of f” (Abramson, 2017, section 3.4). The domain of f °g has the inputs of g that make the outputs
of f.
To find the domain, take the denominators and compare them to zero:
y – 2 = 0 3x – 1 = 0 -6 + 7 = 0 (-∞, 2) (-∞, 1/3) (-∞, 7/6)
f(x) = 4 / (x – 2)
g(x) = 5 / (3x – 1)
f(g(x)) = (12x – 4) / (-6x + 7)
2) Find the inverse of the function
.
State the domains and ranges of both the function and the inverse function in terms of intervals of real numbers.
Make f(x) = y so that it would be easier to keep track of the correct x. Exchange place with x and y. Make y again equal to the new function.
y=4+√x−2 x=4+√y−2 x−4=√y−2 (x−4)2=(√y−2)2
x2−8x+16=y−2 x2−8x+18=y
Answer: The inverse function is
F-1(x)=x2-8x=+18
3)
Go to
www.desmos.com/calculator
and obtain the graph of
, its inverse, and
in the same system of axes. About what pair (a, a) are (11, 7) and (7, 11) reflected about?
f (x)=4+√x−2 Domain: [2, ∞)
f−1(x)=x2−8 x+18 Domain: (-∞, ∞)
If you analyze the graph, the (11, 7) and (7, 11) are reflected about at (9, 9), which is along the g(x) = x function.
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OpenStax. (n.d.). Retrieved December 7, 2021, from https://openstax.org/details/books/algebra-and-trigonometry.