Question:
If the range of the function $f(x) = -3x - 3$ is $\{3, -6, -9, -18\}$, then which one of the following is not in the domain of $f$?
If the range of the function $f(x) = -3x - 3$ is $\{3, -6, -9, -18\}$, then which one of the following is not in the domain of $f$?
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To find domain from a given range, solve the function equation for $x$ using each range value and verify which $x$ values are possible.
Updated On: Jun 6, 2025
- $-1$
- $-2$
- $2$
- $5$
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The Correct Option is A
Solution and Explanation
The function is
\[
f(x) = -3x - 3.
\]
Given range values are $\{3, -6, -9, -18\}$. For each range value $y$, solve for $x$:
\[
y = -3x - 3 \implies x = \frac{-3 - y}{3}.
\]
Calculate $x$ for each $y$:
\[
y = 3: x = \frac{-3 - 3}{3} = \frac{-6}{3} = -2,
\]
\[
y = -6: x = \frac{-3 - (-6)}{3} = \frac{-3 + 6}{3} = \frac{3}{3} = 1,
\]
\[
y = -9: x = \frac{-3 - (-9)}{3} = \frac{-3 + 9}{3} = \frac{6}{3} = 2,
\]
\[
y = -18: x = \frac{-3 - (-18)}{3} = \frac{-3 + 18}{3} = \frac{15}{3} = 5.
\]
Domain values that produce the range are $\{-2, 1, 2, 5\}$. The value $-1$ is not in the domain because it does not produce any $y$ in the range.
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