How do you graph f(x)=4/(x−1)+1 using holes, vertical and horizontal asymptotes, x and y intercepts?

Marcelo Maxwell

Marcelo Maxwell

Answered question

2022-09-23

How do you graph f ( x ) = 4 x - 1 + 1 using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer & Explanation

Simon Zhang

Simon Zhang

Beginner2022-09-24Added 7 answers

We can express f(x) as a single rational function.

f ( x ) = 4 x - 1 + x - 1 x - 1 = x + 3 x - 1

Asymptotes

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve : x - 1 = 0 x = 1 is the asymptote

Horizontal asymptotes occur as

lim x ± , f ( x ) c ( a constant)

divide terms on numerator/denominator by x

f ( x ) = x x + 3 x x x - 1 x = 1 + 3 x 1 - 1 x

as x ± , f ( x ) 1 + 0 1 - 0

y = 1 is the asymptote

Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here, hence there are no holes.

Approaches to vertical asymptote

lim x 1 - = - and lim x 1 + = +

Intercepts

x = 0 f ( 0 ) = 3 - 1 = - 3 y-intercept

y = 0 x + 3 = 0 x = - 3 x-intercept
graph{(x+3)/(x-1) [-10, 10, -5, 5]}

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