inverse of f(x)=(5x+4)/(x+5) — step-by-step solution

⚠ 5x is read as the product 5*x: for a power, write 5^x

1) For a function

\frac{a x + b}{c x + d}

, two inputs u and v with the same output satisfy

(a d - b \cdot c) \cdot (u - v) = 0

after multiplying by the denominators. Here,

5 \cdot 5 - 4

21

≠ 0, so u = v. The function therefore has an inverse on its range

The function domain is

x \in ℝ \ {-5}

Express input

x

in terms of output

y

y = \frac{5 x + 4}{x + 5}

⇥1.1) Let's solve in

ℝ

\frac{5 x + 4}{x + 5} = y

⇥Let's sum the fractions:

\frac{5 x + 4}{x + 5} - y = 0

⇥

\frac{(5 x + 4) - y \cdot (x + 5)}{x + 5} = 0

⇥

\frac{5 x + 4 - y \cdot (x + 5)}{x + 5} = 0

⇥1.2) Let's solve in

ℝ

5 x + 4 - y \cdot (x + 5) = 0

⇥

5 x + 4 - (y x + y \cdot 5) = 0

⇥

5 x

+ 4

- y \cdot x

- y \cdot 5 = 0

⇥

5 x - y x = - 4 + y \cdot 5

⇥

(5 - y) \cdot x = - 4 + y \cdot 5

⇥Solution obtained:

x = \frac{5 y - 4}{5 - y}

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

x + 5 \neq 0

⇥Solution rejected:

x \neq -5

Renaming the output as

x

, the inverse is

\frac{5 x - 4}{5 - x}

, with domain

x \in ℝ \ {5}

▶Inverse function:

\frac{5 x - 4}{5 - x}