Identify the horizontal asymptote of f(x) = quantity 2 x minus 1 over quantity x squared minus 7 x plus 3.
1 answer:
We must recall that a horizontal asymptote is the value/s of y that the given function approaches to but never reaches. To find this in a rational function, we compare the expressions with highest degree in the numerator and denominator. There are three possible outcome when this happens.
1. if the highest degree (highest exponent) in the numerator is bigger than that of the denominator, then there won't be any horizontal asymptote.
2. if the highest degree in the denominator is bigger, then the horizontal symptote would be y = 0.
3. if they have the same highest degree, then we just get the quotient of their coefficient.
Now, going back to our function, we have
From this we can see that the highest degree in the numerator is 1 (from 2x) and 2 (from x²) for the denominator. Clearly, it shows that its denominator has a higher degree. And from our discussion, we can conclude that the horizontal asymptote would be y = 0.
Answer: y = 0
1. if the highest degree (highest exponent) in the numerator is bigger than that of the denominator, then there won't be any horizontal asymptote.
2. if the highest degree in the denominator is bigger, then the horizontal symptote would be y = 0.
3. if they have the same highest degree, then we just get the quotient of their coefficient.
Now, going back to our function, we have
From this we can see that the highest degree in the numerator is 1 (from 2x) and 2 (from x²) for the denominator. Clearly, it shows that its denominator has a higher degree. And from our discussion, we can conclude that the horizontal asymptote would be y = 0.
Answer: y = 0
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#2. First, we factor each polynomial. Then, if any terms on both the top and the bottom of the fraction match, they cancel out. So... we do just that. You end up with:
Notice there's an (x-4) on both top and bottom. So they cancel out. That leaves us with your answer of
#3. We do the same thing as above then multiply and simplify. In the interest of space, I'll cut straight to some simplification.
Now we start cancelling. For the first fraction, there are 3 (x+2)'s on top and 1 on the bottom so we will cancel out the one on the bottom and leave 2 (x+2)'s on top. There are no more polynomials to cancel out so now we multiply across:
10 and 6 share a GCF of 2 so we divide both of those by 2. This leaves us with the final answer of:
#4. This equation introduces division and because of it, we must flip the second fraction to make the division sign into a multiplication symbol. Again for space, I'll flip the fraction and simplify in one step.
Now we do our cancelling. First fraction has (x - 2) in the top and bottom. They're gone. The first fraction has a (x + 4) on the bottom and the second fraction has one on the top. Those will also cancel. This leaves you with:
3 and 6 share a GCF of 3 so we divide both numbers by this. This leaves you with your final answer:
#5. We are adding so we first factor both fractions and see what we need to multiply by to make the denominators the same. I'll do the former first. (10 - x) and (x - 10) are not the same so we multiply the first equation (top and bottom) by (x - 10) and the second equation by (10 - x). Because they will now have the same denominator we can combine them already. This gives us:
Now we FOIL each to expand and then simplify by combining like terms. Again for space, I'm just showing the result of this; you end up with:
Now we factor the top. This gives you 2 (x - 10)'s on top and one on bottom. So we just leave one on the top and cancel the bottom one out. This leaves you with your answer:
#6. Same process for this one so I won't repeat. I'll just show the work.
becomes
which equals
giving you the final answer
#7. For this question we find the least common denominator to make the denominators match. For 5, x, and 2x, the LCD is 10x. So we multiply top and bottom of each fraction by what would make the bottom equal 10x. This rewrites the fraction as:
Simplify to get:
After simplifying again, you end up with your final answer:
Notice there's an (x-4) on both top and bottom. So they cancel out. That leaves us with your answer of
#3. We do the same thing as above then multiply and simplify. In the interest of space, I'll cut straight to some simplification.
Now we start cancelling. For the first fraction, there are 3 (x+2)'s on top and 1 on the bottom so we will cancel out the one on the bottom and leave 2 (x+2)'s on top. There are no more polynomials to cancel out so now we multiply across:
10 and 6 share a GCF of 2 so we divide both of those by 2. This leaves us with the final answer of:
#4. This equation introduces division and because of it, we must flip the second fraction to make the division sign into a multiplication symbol. Again for space, I'll flip the fraction and simplify in one step.
Now we do our cancelling. First fraction has (x - 2) in the top and bottom. They're gone. The first fraction has a (x + 4) on the bottom and the second fraction has one on the top. Those will also cancel. This leaves you with:
3 and 6 share a GCF of 3 so we divide both numbers by this. This leaves you with your final answer:
#5. We are adding so we first factor both fractions and see what we need to multiply by to make the denominators the same. I'll do the former first. (10 - x) and (x - 10) are not the same so we multiply the first equation (top and bottom) by (x - 10) and the second equation by (10 - x). Because they will now have the same denominator we can combine them already. This gives us:
Now we FOIL each to expand and then simplify by combining like terms. Again for space, I'm just showing the result of this; you end up with:
Now we factor the top. This gives you 2 (x - 10)'s on top and one on bottom. So we just leave one on the top and cancel the bottom one out. This leaves you with your answer:
#6. Same process for this one so I won't repeat. I'll just show the work.
#7. For this question we find the least common denominator to make the denominators match. For 5, x, and 2x, the LCD is 10x. So we multiply top and bottom of each fraction by what would make the bottom equal 10x. This rewrites the fraction as:
Simplify to get:
After simplifying again, you end up with your final answer:
Answer:
- One proportion to solve is 8/18 = x/8
- Exact answer as a fraction: x = 32/9
- Approximate answer in decimal form: x = 3.556
In reality, the 5's go on forever, but we have to round off somewhere.
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Explanation:
Break up the triangles as your teacher recommends. See the diagram below as to what I mean.
The smaller triangle has A = 8 as the horizontal side, and B = x as the vertical side
The larger triangle has C = 10+8 = 18 as the horizontal aside, and D = 8 as the vertical side.
We can form the proportion A/C = B/D.
A/C connects the horizontal sides (small/large) while B/D has the vertical sides tied together. The order of division is the same small over large. You could do large over small, but make sure you keep both sides consistent.
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Let's solve for x
A/C = B/D
8/18 = x/8
8*8 = 18*x .... cross multiply
64 = 18x
18x = 64
x = 64/18 .... divide both sides by 18
x = 32/9 exactly in fraction form
x = 3.556 approximately in decimal form
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Another proportion you can solve is A/B = C/D and you should get the same x value. It's the same as A/C = B/D because the B and C swap places. There are many other approaches you could take using different proportions.
