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If f: R → R is defined by f(x) = 2x + 3 and g: R → R is defined by g(x) = x² + 1, then (gof)⁻¹(10) equals

RankGuru · Accepted Answer

1. **Determine Composite Function**: $(g \circ f)(x) = g(f(x)) = g(2x + 3) = (2x + 3)^2 + 1$.
2. **Identify Inverse set**: $(g \circ f)^{-1}(10)$ is the set of all $x$ such that $(g \circ f)(x) = 10$.
3. **Solve Equation**:
   - $(2x + 3)^2 + 1 = 10 \implies (2x + 3)^2 = 9$
   - Taking square roots: $2x + 3 = 3$ or $2x + 3 = -3$.
4. **Evaluate $x$**:
   - $2x = 0 \implies x = 0$
   - $2x = -6 \implies x = -3$
5. **Result**: The pre-image set is **$\{0, -3\}$**.

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