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The curves $f(x)=3^{x}$ and $g(x)=\log_3 x$ are drawn on the same axes. Their relationship is that:
Athey meet only at the origin
Bthey are reflections in the $x$-axis
Cthey are reflections in the line $y=x$
Dboth increase at the same rate
Answer & Solution
Correct answer: C. they are reflections in the line $y=x$
1. $g(x)=\log_3 x$ is by definition the inverse of $f(x)=3^{x}$.
2. The graph of a function and its inverse are mirror images in the line $y=x$.
3. This shows in the intercepts: $f$ passes through $(0,1)$ and $g$ through $(1,0)$, a swapped pair.
4. Reflection in the $x$-axis would send $3^{x}$ to $-3^{x}$, which is negative everywhere, so that option fails.
5. Saying both simply 'increase' describes them without capturing the symmetry.
_Source: NECTA ACSEE 2023 Advanced Mathematics 142/1, Question 6: Functions_
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