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VICMath MethodsQuick questions
Unit 4
Quick questions on Hybrid functions and inverse functions: VCE Math Methods Unit 4
15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is continuity at a join?Show answer
A hybrid function is continuous at the join point $x = a$ if the left-hand value and right-hand value of $f$ agree at $a$:
What is differentiability at a join?Show answer
A hybrid function is differentiable at the join point $x = a$ if it is continuous at $a$ AND the left and right derivatives agree:
What is two-constant problems?Show answer
Many Paper 1 hybrid questions give a hybrid function with two unknown constants in one piece, and ask to find the constants such that the function is continuous and differentiable at a specified join.
What is finding $f^{-1}$ algebraically?Show answer
To find the inverse of $y = f(x)$:
What is domain and range swap?Show answer
If $f: A \to B$ is one-to-one, then $f^{-1}: B \to A$ (with the swapped roles).
What is graphical interpretation?Show answer
The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.
What is worked example. Linear function?Show answer
$f(x) = 3x - 7$ on $\mathbb{R}$.
What is worked example. Exponential?Show answer
So $f^{-1}(x) = \ln(x)$, with domain $(0, \infty)$ and range $\mathbb{R}$.
What is worked example. Quadratic requiring restriction?Show answer
$f(x) = x^2$ on $\mathbb{R}$ has no inverse (not one-to-one; $f(-2) = f(2) = 4$).
What is confusing $f^{-1} $ with $1 / f $?Show answer
The notation $f^{-1}$ means functional inverse, not reciprocal. $f^{-1}(x) \neq 1 / f(x)$ in general.
What is forgetting domain restriction?Show answer
Asking for the inverse of $x^2$ on $\mathbb{R}$ has no answer. Restrict to make $f$ one-to-one before inverting.
What is wrong root sign?Show answer
When solving $y = x^2$ for $x$, the answer is $x = \pm \sqrt{y}$. The correct sign depends on the domain restriction.
What is domain and range not swapped?Show answer
The inverse's domain is the original's range, and vice versa. Stating only the original domain misses the swap.
What is continuity matched but differentiability ignored?Show answer
"Continuous and differentiable" requires both conditions. If only continuity is checked, the function may have a kink.
What is differentiability checked without first checking continuity?Show answer
A function that is not continuous at a join cannot be differentiable there. Check continuity first.