SOLUTION: Let \[f(x) = \frac{3x - 7}{x + 1}.\]Find the range of $f$. Give your answer as an interval.

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Question 1088255: Let
\[f(x) = \frac{3x - 7}{x + 1}.\]Find the range of $f$. Give your answer as an interval.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The function f%28x%29+=+%283x-7%29%2F%28x%2B1%29 has the horizontal asymptote %283x%29%2Fx+=+3%2F1+=+3.

You divide the leading terms where x+%3C%3E+0.

As x gets larger and larger, f(x) will approach f(x) = 3.

What this means is that any value but y = 3 is possible for the range.

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If we have y+=+%283x-7%29%2F%28x%2B1%29 and we swapped x and y, then we'd have x+=+%283y-7%29%2F%28y%2B1%29

Let's solve for y

x+=+%283y-7%29%2F%28y%2B1%29

x%28y%2B1%29+=+3y-7

xy%2Bx+=+3y-7

xy-3y+=+-7-x

y%28x-3%29+=+-7-x

y+=+%28-7-x%29%2F%28x-3%29

which is the inverse function. The denominator of this inverse, x-3, indicates that 3 is not in the domain of the inverse. This confirms that 3 is not in the range of the original function.

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So the range is the set of all y values such that y is a real number but y cannot equal 3.

In set builder notation, we'd say

In interval notation, we'd write

The big U means "union" to essentially "glue" the two intervals together.

Here's what the graph looks like. The blue dashed line is the horizontal asymptote y = 3 (it goes through the two points (0,3) and (1,3))

The green curve approaches the dashed line but it will never cross it or touch it. So this shows us that any output but y = 3 is possible.