What is the equation of the function y=2/x translated 3 units to the left and 4 units up?
A. y = 2/x+3 + 4
B. y = 2/x-3 - 4
The correct answer is: y = 2/(x+3) + 4
can you show how you got that
Sure! To translate a function 3 units to the left and 4 units up, we need to add 3 to the x-coordinate and 4 to the y-coordinate of each point on the original function.
Starting with the original function y = 2/x:
1. To shift 3 units to the left, we substitute x+3 for x:
y = 2/(x+3)
2. To shift 4 units up, we add 4 to the y-coordinate:
y + 4 = 2/(x+3)
3. Solving for y, we get:
y = 2/(x+3) + 4
Therefore, the final equation of the function with the requested translation is: y = 2/(x+3) + 4.
Simplify the complex fraction
n-4/n^2-2n-15/n+1/n+3
A. n-4/(n-5)(n+1)
B. n-4/(n+3)
To simplify the complex fraction, we need to first factor all the denominators, and then multiply by the reciprocal of the complex fraction's divisor. Let's break it down step by step:
n - 4 / (n^2 - 2n -15) / (n + 1) / (n + 3)
= (n - 4) / [(n - 5)(n + 3)] * [(n + 1) / (n + 1)]
Note that we have separated the expression into one fraction over another by turning the division operation into multiplication by taking the reciprocal (flipping the second fraction and multiplying):
[(n - 4) / (n^2 - 2n - 15)] * [(n + 1) / (n + 3)]
= [(n - 4) / ((n - 5)(n + 3))] * [(n + 1) / (n + 3)]
The factors of the denominator are now in their simplest form. We can see that there is a common factor of (n + 3) in both the numerator and the denominator of the complex fraction. We can cancel this factor:
= (n - 4) / (n - 5)(n + 1)
Therefore, the simplified form of the given complex fraction is:
(n - 4) / (n - 5)(n + 1)
Hence, option (A) is the correct answer.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
The sum of the reciprocals of two consecutive even integers is
9/40. This can be represented by the equation shown.
1/x + 1/x+2 = 9/40
1/x + 1/(x+2) = 9/40
To solve for x, we need to get rid of the denominators on both sides of the equation. We can do this by finding a common denominator of (x)(x+2) for the left-hand side of the equation:
[(x+2) + x] / (x)(x+2) = 9/40
Simplifying the numerator on the left-hand side:
2x + 2 / (x)(x+2) = 9/40
Next, we can cross-multiply the fractions:
40(2x + 2) = 9(x)(x+2)
Simplifying both sides of the equation:
80x + 80 = 9x^2 + 18x
9x^2 - 62x - 80 = 0
We can use the quadratic formula to solve for x:
x = [62 ± sqrt(62^2 - 4(9)(-80))] / (2*9)
x = [62 ± sqrt(3844 + 2880)] / 18
x = [62 ± sqrt(6724)] / 18
x = [62 ± 82] / 18
x = 3.44 or x = -5.78
Since we are looking for consecutive even integers, we round 3.44 down to 2 and eliminate -5.78 as an extraneous solution.
Therefore, the two consecutive even integers are 2 and 4.
As a check, we can plug x = 2 into the original equation:
1/x + 1/(x+2) = 1/2 + 1/4 = 2/4 + 1/4 = 3/4
3/4 does not equal 9/40, so the solution is not valid. However, if we plug in x = 4, we get:
1/x + 1/(x+2) = 1/4 + 1/6 = 3/12 + 2/12 = 5/12
5/12 can be simplified by multiplying both sides of the equation by (12/5), which gives:
x = 4, x + 2 = 6, and 5/12 * (12/5) = 1
So, our solution is x = 4, which means that the consecutive even integers are 2 and 4.