Rewrite as y = (5x+2)/(x-1)
y - intercept is when x=0. Plug in 0 for x and solve.
y = (5(0) + 2)/(0-1) = 2/-1 = -2 is the x-intercept.
To find the x-intercept of a rational function find x-values that make the function = 0. x-intercept is when y = 0. You only look at the numerator.
5x+2=0
5x=-2
x = -2/5
Plug that value back into the equation to ensure it makes the function = 0.
5(2/5)+2 / 2/5 - 1 = -2+2 / (2/5 - 1) = 0 / (2/5 - 1) = 0. So x=-2/5 is the x-intercept.
Vertical asymptote of a rational function is a value that makes the denominator = 0, or places a negative number in a square root.
In this case x-1=0 when x=1. So x=1 is a vertical asymptote.
For horizontal asymptote look at the biggest power of x in the numerator and denominator. If the biggest power in the numerator is bigger then the function has no horizontal asymptote. If the power in the denominator is bigger then the function has an asymptote at y=0. If the powers are the same, as they are here, then the horizontal asymptote is the coefficient of the numerator largest power term divided by the coefficient of the denominator largest term. In this case that's 5/1 = 5.
There are a variety of methods to find the range. The easiest and first to try is graph the equation. If you graph this equation you see the y values approach positive and negative infinity as the graph approaches the vertical asymptote. The range is therefore -infinity to +infinity. You could also plug in x-values that are close to the asymptote from both sides. That will also show that y approaches - infinity and + infinity.
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