Would someone kindly verify these answers~

Thank you!
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1.where does the reciprocal function of f(x)=3-x increase?

a){XER|x>3}
b){XER|x<3}
c){XER|x≠3}
d){XER}

Answer: C

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2. What is the domain of the reciprocal function of f(x)=5x+1?

a) {XER|x≥-1/5}
b) {XER|x≤-1/5}
c) {XER|x≠-1/5
d) {XER}

Answer: C
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3. State the range of f(x)= 1/3x+4

a) [YER|y>-4/3]
b) [YER|y≠-4/3]
c) [YER|y>0]
d) [YER|y≠0]

Answer B
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1. correct

2. correct
3. did you mean fx) = 1/(3x+4) ?
if so, then the range is d)

For 3, its the reciprocal of 3x+4

There is no brackets.

3x+4=0
3x=-4
x≠-4/3

So b?

@ Luckylin and Reiny, yes, no brackets

If there are no brackets, then the equation is simply the straight line

y = (1/3)x + 4
the range of that line is y ∊R, which is none of the choices, so you must be mistaken

I bet you have a numerator of 1 and a denominator of 3x+4
which has to be written as 1/(3x+4) and I stand by my answer of d)

1. "Where does the reciprocal function of f(x)=3-x increase? Well, first let's all agree that math is very confusing. Now, the reciprocal function of 3-x is just 1/(3-x). So, where does this function increase? Hmm... let me think. You know what? Let's just go with option C. It seems like a nice, safe answer. Plus, it has that fancy '≠' symbol, which always makes me feel smart."

2. "Ah, the domain of the reciprocal function of f(x)=5x+1. Sounds thrilling, doesn't it? Now, to figure this out, we need to keep in mind that the reciprocal of a number is just 1 divided by that number. So, the reciprocal function of 5x+1 would be 1/(5x+1). And where is this function defined? Well, I think we can confidently say it's not defined when x is equal to -1/5. So, let's go with option C. It's always fun to exclude a specific number from the party!"

3. "The range of f(x)=1/3x+4. Oh boy, we're diving into fractions now! Don't worry, I'll try to keep it simple. The range is basically the set of all possible y-values that the function can take. So, let's analyze this function. As x gets really big, the whole fraction gets really small. And as x approaches negative infinity, the fraction becomes really negative. So, we can safely say that y can be any value except -4/3. Therefore, option B seems like the way to go. Yay, avoiding fractions just got a little bit easier!"

To verify the answers provided, let's go through the process of finding the solutions step by step.

1. For the reciprocal function of f(x) = 3 - x to increase, the original function must be positive. In other words, we need to find the value(s) of x for which f(x) > 0. To do this, we set the original function f(x) equal to 0 and solve for x:

3 - x = 0
x = 3

Since x ≠ 3 (as dividing by zero is undefined), the reciprocal function will be positive (or increase) for all values of x except when x = 3. Therefore, the correct answer is option c) {XER|x≠3}.

2. To determine the domain of the reciprocal function of f(x) = 5x + 1, we need to find the values of x for which the original function is defined. The original function can be any real number. However, when finding the reciprocal, we need to avoid dividing by zero. So, we set the original function equal to zero and solve for x:

5x + 1 = 0
5x = -1
x = -1/5

Since x ≠ -1/5 (as dividing by zero is undefined), the domain of the reciprocal function will be all real numbers except x = -1/5. Therefore, the correct answer is option c) {XER|x≠-1/5}.

3. In order to find the range of the function f(x) = 1/3x + 4, we need to consider the possible values of y (the output) when x (the input) varies. Since the coefficient of x is positive (1/3), the function will have positive values for y, except when x approaches negative infinity.

To find the specific value that causes the function to equal zero, we set the function equal to zero and solve for x:

1/3x + 4 = 0
1/3x = -4
x = -12

Since y can be any value greater than -4/3 (as x approaches negative infinity), the correct answer is option a) [YER|y>-4/3].

By going through the steps, we can verify that the correct answers are indeed:

1. Answer: c) {XER|x≠3}
2. Answer: c) {XER|x≠-1/5}
3. Answer: a) [YER|y>-4/3]