A decreasing function g satisfies g(4) = 6 and g'(4) = -2. Which of the following statements about the inverse of g must be true? a. (g^-1)'(6) = -2
b. (g^-1)'(-2) = 4
c. (g^-1)'(6) = -2
d. (g^-1)'(6) = -1/2

The statement that is true about the inverse of g from the given options is;

Option D; (g ⁻¹)'(6) = -1/2

The theorem for inverse functions is one that is stated by the relationship that if f(x) = y, then the derivative of the inverse function is given as;

(f ⁻¹)'(y) = 1/f'(x) = 1/f'(f ⁻¹(y))

From the question, we see that; g(4) = 6 and g'(4) = -2. Thus, from the theorem, we can see that;

(g ⁻¹)'(6) = 1/g'(4)

Plugging in -2 for g'(4) gives;

(g ⁻¹)'(6) = -1/2

Looking at the options, the only one that corresponds with our answer is Option D.

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The inverse function theorem (terms and conditions apply) says that, if f(a) = b, then

(f ⁻¹)'(b) = 1 / f '(a) = 1 / f '(f ⁻¹(b))

We're given that g (4) = 6, so from the theorem it follows that

(g ⁻¹)'(6) = 1 / g '(4) = -1/2

so only D must be true.

B might also be true, but we don't have enough information to determine that.

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step-by-step explanation:

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: