Studying a function
So far each section of this unit has worked on one property of a function at a time: its domain, its symmetry, its growth, its long-term behaviour. In Bachillerato you will be asked to bring all of these properties together into a single analysis, carried out systematically and in a fixed order, so that you can sketch the graph without having to tabulate dozens of points. That procedure is called "studying a function", and it is the one you will practise in this section.
The procedure, step by step
Always following the same order stops you forgetting properties and lets you reuse each result in the next step: for example, if the function is odd, it is enough to study the monotonicity on the positive half of the domain.
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, intercepts and symmetry
Find the axis intercepts of the function $f(x) = \frac{3x - 6}{x + 2}$.
Find the axis intercepts of the function $f(x) = \frac{3x - 6}{x + 2}$.
Determine analytically whether the function $f(x) = x^4 - 5x^2 + 3$ has even symmetry, odd symmetry or no symmetry.
Determine analytically whether the function $f(x) = x^4 - 5x^2 + 3$ has even symmetry, odd symmetry or no symmetry.
Find the domain of definition of the irrational function $f(x) = \sqrt{2x - 8}$.
Find the domain of definition of the irrational function $f(x) = \sqrt{2x - 8}$.
Find the domain of definition of the rational function $f(x) = \frac{x - 1}{x^2 - 4x - 5}$.
Find the domain of definition of the rational function $f(x) = \frac{x - 1}{x^2 - 4x - 5}$.
Monotonicity, extrema 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 $[-1, 2]$.
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.