Could someone help/verify these answers~

Thanky you!

1.What is the range of the reciprocal function of f(x)=10-x^2?

a) {YER|y≥1/10}
b) {YER|y≥10}
c) {YER|1/10≤y<0}
d) YER|y<0 or y≥1/10}

Answer: D?
----------------------------------------
2. It takes Franc 2 hours longer than jane to carpet a certain type of room. Together they can carpet that type of room in 1 7/8 hours. How long would it take for Frank to do the job alone?

a) 1 1/4 hours
b) 3 3/4 hours
c) 3 hours
d) 5 hours
---------------------------------------
4. Solve 4x/x-2 = 3x-2/x-2 for x.

a) x= -1/2
b) x= -2, 2
c) x= -2
d) x=2

Answer: B
---------------------------------------
5. How many solutions does this equation (x+5)/(x-3)-1/x = 4/(x^2-3x) have?

a) 0
b) 1
c) 2
d) 3

Answer: C
---------------------------------------
6. The inequality 3x-2<(x+4)/(x-2) is equivalent to which of the following?

a) 3x^2 - 9x + 8<0
b) (3x^2-9x+8)/(x-2)<0
c) 3x^2-9x>0
d) (3x^2 - 9x)/(x-2)<0

Answer: D

---------------------------------------

1.

y = 10 - x^2
x = sqrt(10-y)
domain: y <= 10
I don't see that as a choice.

2.
1/f + 1/j = 15/8
but f = j+2
1/(j+2) + 1/j = 15/8
8(j + j+2) = 15j(j+2)
16j + 16 = 15j^2 + 30j
15j^2 + 14j - 16 = 0
(3j-2)(5j+8) = 0
j = 2/3
so, f = 2 2/3
I don't see that as a choice.

4.
The answer is C, not B because 1/(x-2) is not defined.

5. C correct

6. D correct

For number 4, I got this answer after calculating...

4x/(x-2)=(3x-2)/(x-2)

4x(x-2)=(3x-2)(x-2)
4x^2 - 8x = 3x^2 - 6x - 2x + 4
4x^2-3x^2-8x+6x+2x-4=0
x^2 - 4 =0

(x-2)(x+2)=0

x= 2, -2

Am I doing something wrong?

Yes. 2 is not allowed as a solution, since putting it into the original equation gives

4x/(x-2)=(3x-2)/(x-2)
48/0 = 10/0

make that

4/0 = 10/0

1. To find the range of the reciprocal function of f(x)=10-x^2, we need to find the range of the original function and then take the reciprocal of each value.

First, let's find the range of f(x)=10-x^2. This is a quadratic function with a negative coefficient in front of the x^2 term. The graph of such a function opens downwards, meaning the maximum value occurs at the vertex. To find the vertex, we can use the formula x = -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = -1 and b = 0, so the vertex occurs at x = 0.

Now, let's substitute this x value back into the original function to find the corresponding y value. f(0) = 10 - (0^2) = 10.

So, the vertex of the function is (0, 10), which means the maximum value of the function is 10.

Since the function is decreasing on both sides of the vertex, the range of the function is (-∞, 10]. To find the range of the reciprocal function, we take the reciprocal of each value in the range.

The reciprocal of y is 1/y. So the range of the reciprocal function is [1/∞, 1/10] = (-∞, 1/10].

Therefore, the correct answer is (d) YER|y<0 or y≥1/10.

2. Let's assume that Jane takes x hours to carpet the room. Since Fran takes 2 hours longer than Jane, we can say that Fran takes x+2 hours to carpet the room.

Together, they can carpet the room in 1 7/8 hours.

To find their combined work rate, we add their individual work rates together. The formula for work rate is 1/time.

Jane's work rate is 1/x and Fran's work rate is 1/(x+2).

Their combined work rate is (1/x) + (1/(x+2)).

We are given that their combined work rate is 1/1 7/8. We can convert 1 7/8 to an improper fraction: (8 + 7)/8 = 15/8.

So the equation becomes:

1/x + 1/(x+2) = 1/(15/8)

To solve this equation, we can multiply each term by the least common denominator (24x(x+2)) to eliminate the fractions:

24(x+2) + 24x = 3x(x+2)

Simplifying further:

24x + 48 + 24x = 3x^2 + 6x

48x + 48 = 3x^2 + 6x

Rearranging:

3x^2 - 42x - 48 = 0

Factoring out a common factor of 3:

3(x^2 - 14x - 16) = 0

Factoring the quadratic expression:

3(x - 16)(x + 2) = 0

Setting each factor equal to zero:

x - 16 = 0 or x + 2 = 0

Solving for x:

x = 16 or x = -2

Since we are looking for a positive time, we discard the negative solution x = -2.

Therefore, it would take Frank 16 hours to do the job alone.

So the correct answer is (d) 5 hours.

4. To solve the equation 4x/(x-2) = 3x-2/(x-2) for x, we can start by multiplying both sides of the equation by (x-2) to eliminate the denominators:

4x = 3x - 2

Next, we can subtract 3x from both sides of the equation:

4x - 3x = -2

Simplifying further:

x = -2

So the correct answer is (a) x= -1/2.

5. Let's simplify the equation (x+5)/(x-3) - 1/x = 4/(x^2-3x) to determine the number of solutions it has.

To do this, let's find the least common denominator (LCD) of the fractions involved, which is (x-3)(x). By multiplying every term in the equation by the LCD, we can eliminate the fractions:

(x-3)(x)((x+5)/(x-3)) - (x-3)(x)(1/x) = (x-3)(x)(4/(x^2-3x))

Simplifying:

(x+5)(x) - (x-3)(x-3) = 4

Expanding the brackets:

x^2 + 5x - x^2 - 5x + 9x - 9 = 4

Simplifying further:

9x - 9 = 4

Adding 9 to both sides:

9x = 13

Dividing both sides by 9:

x = 13/9

So the equation has exactly one solution.

Therefore, the correct answer is (b) 1.

6. To determine which inequality is equivalent to 3x-2<(x+4)/(x-2), we can start by multiplying both sides of the inequality by (x-2) to eliminate the fraction:

3x(x-2) - 2(x-2) < x + 4

Expanding and simplifying:

3x^2 - 6x - 2x + 4 < x + 4

Combining like terms:

3x^2 - 8x < x

Moving all terms to one side of the inequality:

3x^2 - 8x - x < 0

Simplifying further:

3x^2 - 9x < 0

Factoring a common factor of 3x:

3x(x - 3) < 0

To determine the sign of this inequality, we can examine the sign of each factor.

When x < 0, both factors are negative, and the inequality holds true.

When x > 3, both factors are positive, and the inequality does not hold true.

When x = 0 or x = 3, the inequality is not defined due to division by zero from the original fraction.

Therefore, the correct answer is (d) (3x^2 - 9x)/(x-2) < 0