Victorian VCE · VCE Mathematical Methods

Functions, Relations and Composite Transformations

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Let $f (x) = 4 - x^{2}$ and $g (x) = 2 x - 1$ . The maximal domain of the composite function $f (g (x))$ is:
The transformation $T : R^{2} \to R^{2}$ maps $(x, y)$ to $(x^{'}, y^{'})$ where $x^{'} = 3 x - 1$ and $y^{'} = - 2 y + 4$ . If the image curve has equation $y^{'} = (x^{'})^{2}$ , what was the equation of the or…
Find the rule and domain of the inverse function $f^{- 1} (x)$ for $f : [2, \infty) \to R, f (x) = (x - 2)^{2} + 3$ .
The graph of $y = \frac{a}{x - b} + c$ has asymptotes at $x = 3$ and $y = - 2$ , and passes through the origin $(0, 0)$ . What is the value of $a$ ?
Determine the period and amplitude of the function $h (t) = 5 - 3 cos (\frac{π t}{4})$ .
For the function $f (x) = x^{3} - 3 k x^{2} + 9 x - 1$ to have strictly no stationary points, the real parameter $k$ must satisfy:
Solve $2^{x + 1} = 3 \cdot 2^{x} - 4$ for $x$ .
How many solutions does $2 cos (2 x) = 1$ have for $x \in [0, 2 π]$ ?
The graph of $y = f (x)$ is dilated by a factor of $\frac{1}{2}$ from the $y$ -axis, and then translated 3 units to the left. The point $(4, 6)$ on the original graph moves to:
For $f (x) = x^{3} + x$ , how many points of intersection are there between the graphs of $y = f (x)$ and $y = f^{- 1} (x)$ ?
What is the range of the function $f (x) = 2 sin (3 x) + 1$ ?
The function $f (x) = 3 x - 5$ has an inverse. Its rule is $f^{- 1} (x) =$
The graph of $y = x$ is reflected in the $y$ -axis and then translated 4 units up. Its equation is:
Solve $lo g_{2} (x) + lo g_{2} (x - 2) = 3$ for $x$ .
How many solutions does $tan (x) = 1$ have for $x \in [0, 3 π]$ ?
Let $f (x) = x^{2} + 1$ for $x \geq 0$ and $g (x) = x - 1$ for $x \geq 1$ . The composite $f (g (x))$ has rule and domain:
The sum of all solutions of $2 sin^{2} (x) = 1 + cos (x)$ for $x \in [0, 2 π]$ is:
Let $f : (- \infty, 1] \to R$ , $f (x) = x^{2} - 2 x + 5$ . The inverse function has rule and domain:
What is the period of $y = 4 cos (\frac{x}{3}) - 1$ ?
For $f (x) = \frac{2 x + 1}{x - 3}$ , $x \neq = 3$ , the inverse function is $f^{- 1} (x) =$

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Functions, Relations and Composite Transformations

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