intercepts of f(x)=(9-3x)/(x-5) — step-by-step solution

On the horizontal axis, y = 0: solve

\frac{9 - 3 x}{x - 5} = 0

1) Let's solve in

ℝ

\frac{9 - 3 x}{x - 5} = 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

9 - 3 x = 0

⇥

3 x = 9

⇥ⓘ Both sides are divided by

3

⇥

x = \frac{9}{3}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

3

, their greatest common divisor:

\frac{9}{3}

3

⇥Candidates, pending the domain check:

x = 3

⇥1.2) To prevent the denominator from being zero, let's solve:

x - 5 \neq 0

⇥Domain condition:

x \neq 5

For the expression

\frac{9 - 3 x}{x - 5}

to be defined, we must have:

x - 5 \neq 0

Domain constraints:

x \neq 5

The graph crosses the horizontal axis at

(3, 0)

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

⇥

\frac{9 - 3 \cdot 0}{0 - 5}

⇥ⓘ A product with a zero factor is zero:

-3 \cdot 0

0

■Adding 0 changes nothing:

0 - 5

-5

■The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{9 + 0}{-5}

- \frac{9 + 0}{5}

⇥

- \frac{9 + 0}{5}

⇥ⓘ Adding 0 changes nothing:

9 + 0

9

⇥

- \frac{9}{5}

▶Axis intercepts:

{(3, 0), (0, - \frac{9}{5})}