inverse of f(x)=1.7sqrt(-(x-7.35))-3.6 — step-by-step solution | Explain MAths

inverse of f(x)=1.7sqrt(-(x-7.35))-3.6

The function is strictly monotone on

x \leq 7.35

: it has an inverse on its range

y \geq -3.6

Isolate the square root:

\frac{y + 3.6}{1.7}

≥ 0. Under this condition, squaring gives

- x + 7.35 = {(\frac{y + 3.6}{1.7})}^{2}

⇥Let's solve in

ℝ

:

- x + 7.35 = {(\frac{y + 3.6}{1.7})}^{2}

⇥

7.35 - {(\frac{y + 3.6}{1.7})}^{2} = x

⇥

7.35 - \frac{{(y + 3.6)}^{2}}{{1.7}^{2}} = x

⇥

7.35 - \frac{{(y + 3.6)}^{2}}{2.89} = x

⇥

7.35 - \frac{(y + 3.6) \cdot (y + 3.6)}{2.89} = x

⇥

7.35 - \frac{y \cdot y + y \cdot 3.6 + 3.6 y + 3.6 \cdot 3.6}{2.89} = x

⇥ℹ

y \cdot y

=

y^{2}

‖

y \cdot 3.6 + 3.6 y

=

7.2 y

‖

3.6 \cdot 3.6

=

12.96

⇥

7.35 - \frac{y^{2} + 7.2 y + 12.96}{2.89} = x

⇥

\frac{2.89 \cdot 7.35 - (y^{2} + 7.2 y + 12.96)}{2.89} = x

⇥ℹ

- (y^{2} + 7.2 y + 12.96)

=

- y^{2} - 7.2 y - 12.96

‖

2.89 \cdot 7.35

=

21.2415

⇥

\frac{21.2415 - y^{2} - 7.2 y - 12.96}{2.89} = x

⇥

\frac{8.2815 - y^{2} - 7.2 y}{2.89} = x

⇥Solution found:

x = \frac{- y^{2} - 7.2 y + 8.2815}{2.89}

Renaming the output as

x

, the inverse is

\frac{- x^{2} - 7.2 x + 8.2815}{2.89}

, with domain

x \geq -3.6

▶Inverse function:

\frac{- x^{2} - 7.2 x + 8.2815}{2.89}