The inverse of the function $f(x) = \dfrac{2 + 3x}{x + 5}$, by taking $f(x)$ as y, is __________.
(A) $\dfrac{2 + 5y}{y + 3}$
(B) $\dfrac{2 - 5y}{y + 3}$
(C) $\dfrac{2 - 5y}{y - 3}$
(D) $\dfrac{2 + 5y}{y - 3}$

Question

Accepted Answer

> [!TLDR]
> Solve $y = \dfrac{2+3x}{x+5}$ for $x$: $x = \dfrac{2-5y}{y-3}$.

### Step 1 — Cross-multiply
$$y(x+5) = 2 + 3x \;\Rightarrow\; yx + 5y = 2 + 3x$$

### Step 2 — Collect the x terms
$$yx - 3x = 2 - 5y \;\Rightarrow\; x(y-3) = 2 - 5y$$

### Step 3 — Isolate x
$$x = \frac{2 - 5y}{y - 3}$$

This is $f^{-1}(y)$.

**Why the other options are wrong:** (D) $\frac{2+5y}{y-3}$ has a sign error on $5y$; (A) $\frac{2+5y}{y+3}$ mishandles both signs; (B) $\frac{2-5y}{y+3}$ has the denominator sign wrong. …

Step 1 — Cross-multiply

Step 2 — Collect the x terms

Step 3 — Isolate x

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