Studying a function
A bus company counts how many people are on one route as the day goes on: busy at 8:30, quiet at 11:00, busy again at 17:00 (the numbers are made up). To describe a graph like that fully you look at its properties one at a time, in a fixed order. That procedure is called "analysing a function", and you will practise it in this section. At the end we will see how it describes the bus.
The procedure, step by step
Always following the same order stops you forgetting properties and lets you reuse each result in the next step.
Domain and range from a graph
The domain is the set of all $x$-values the graph covers; the range is the set of all $y$-values it reaches. A filled dot means the end point is included (use a square bracket); an open dot means it is not (use a round bracket).
Based on a function whose graph consists of two continuous sections: The first section goes from the point $(-3, 2)$ (included) to the point $(1, 4)$ (not included), and…
Based on a function whose graph consists of two continuous sections:
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The first section goes from the point $(-3, 2)$ (included) to the point $(1, 4)$ (not included), and has a relative maximum at $(-1, 5)$.
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The second section goes from the point $(1, -1)$ (included) to the point $(8, 3)$ (included), and has a relative minimum at $(4, -2)$.
Determine its domain and its range.
Domain and intercepts
Based on a function whose graph consists of two continuous sections: The first section goes from the point $(-3, 2)$ (included) to the point $(1, 4)$ (not included), and…
Based on a function whose graph consists of two continuous sections:
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The first section goes from the point $(-3, 2)$ (included) to the point $(1, 4)$ (not included), and has a relative maximum at $(-1, 5)$.
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The second section goes from the point $(1, -1)$ (included) to the point $(8, 3)$ (included), and has a relative minimum at $(4, -2)$.
Determine its domain and its range.
Turning points and rate of change
Calculate the average rate of change (ARC) of the function $f(x) = 2x^2 - 3x$ on the interval $[-1, 2]$.
Calculate the average rate of change (ARC) of the function $f(x) = 2x^2 - 3x$ on the interval $[-1, 2]$.
The variation in the water level of a reservoir $v$ (in metres relative to its average level) as a function of the month $x$ of the year is given by the function…
The variation in the water level of a reservoir $v$ (in metres relative to its average level) as a function of the month $x$ of the year is given by the function $v(x) = -0.5x^2 + 10x - 18$. Determine in which month the reservoir level is highest, what this maximum difference from the average level is, and in which months the reservoir is at exactly the average level ($v = 0$).
From the following graph, state the intervals of monotonicity (increasing and decreasing) and the coordinates of its local extrema.
From the following graph, state the intervals of monotonicity (increasing and decreasing) and the coordinates of its local extrema.
Study the continuity of the following piecewise function at the point $x = 1$: $f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 1 \\ 4 - x & \text{if } x > 1 \end{cases}$
Study the continuity of the following piecewise function at the point $x = 1$:
$f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 1 \\ 4 - x & \text{if } x > 1 \end{cases}$
The position of an object moving in a straight line is given by the function $e(t) = 5t^2 + 2t$, where $e$ is the distance in metres and $t$ is the time in seconds.…
The position of an object moving in a straight line is given by the function $e(t) = 5t^2 + 2t$, where $e$ is the distance in metres and $t$ is the time in seconds. Calculate the average speed (average rate of change) on the intervals $[1, 3]$ and $[3, 5]$, and state in which interval the object moved faster.
Long-term behaviour and asymptotes
Identify the equations of the vertical asymptote and the horizontal asymptote of the function $f(x) = \frac{5}{x - 3} + 2$.
Identify the equations of the vertical asymptote and the horizontal asymptote of the function $f(x) = \frac{5}{x - 3} + 2$.
Determine the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{3x + 1}{x - 2}$.
Determine the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{3x + 1}{x - 2}$.
A complete example: the profile of a function
Describe the behaviour of the function $f(x) = \frac{2x^2}{x^2 + 1}$ by analysing its domain, its axis intercepts, its symmetry, and its horizontal and vertical…
Describe the behaviour of the function $f(x) = \frac{2x^2}{x^2 + 1}$ by analysing its domain, its axis intercepts, its symmetry, and its horizontal and vertical asymptotes.
Given the following piecewise function, study its domain, its continuity, its axis intercepts and its intervals of monotonicity:…
Given the following piecewise function, study its domain, its continuity, its axis intercepts and its intervals of monotonicity:
$f(x) = \begin{cases} x^2 - 4 & \text{if } x < 0 \\ 2x - 4 & \text{if } 0 \leq x < 4 \\ 4 & \text{if } x \geq 4 \end{cases}$
Prove analytically whether the function $f(x) = \frac{x^3}{x^2 - 4}$ has even symmetry, odd symmetry or neither. Then, calculate its axis intercepts and determine the…
Prove analytically whether the function $f(x) = \frac{x^3}{x^2 - 4}$ has even symmetry, odd symmetry or neither. Then, calculate its axis intercepts and determine the equations of its vertical and horizontal asymptotes.