intercepts of f(x)=(x-3)/(x-4) — step-by-step solution

On the horizontal axis, y = 0: solve

\frac{x - 3}{x - 4} = 0

1) Let's solve in

ℝ

\frac{x - 3}{x - 4} = 0

A quotient is zero if and only if its numerator is zero and its denominator is not: we solve the numerator equal to 0

⇥1.1) Let's solve in

ℝ

x - 3 = 0

⇥Candidates, pending the domain check:

x = 3

⇥1.2) To avoid a zero denominator, let's solve:

x - 4 \neq 0

⇥Domain condition:

x \neq 4

For the expression

\frac{x - 3}{x - 4}

to exist, you must check:

x - 4 \neq 0

Domain constraints:

x \neq 4

The graph crosses the horizontal axis at

(3, 0)

On the vertical axis, x = 0: calculate f(0)

⇥

\frac{0 - 3}{0 - 4}

⇥ⓘ Adding 0 changes nothing:

0 - 3

-3

0 - 4

-4

■The minus signs of the numerator and the denominator cancel out:

\frac{-3}{-4}

\frac{3}{4}

⇥

\frac{3}{4}

▶Axis intercepts:

{(3, 0), (0, \frac{3}{4})}