Domain, range and image of a function
You already know that the domain is the set of values of $x$ where the function exists. Now it is time to be precise: each type of function has its own trap (a fraction whose denominator becomes zero, the root of a negative number, the logarithm of zero), and you need to be able to spot it without having to draw anything.
Polynomial and rational functions
When the denominator is quadratic, you have to solve a quadratic equation to find all the excluded points (there can be none, one or two).
Determine the domain of definition of the following polynomial function: $f(x) = -4x^5 + 7x^3 - x + 12$
Determine the domain of definition of the following polynomial function:
$f(x) = -4x^5 + 7x^3 - x + 12$
Find the domain of the following rational function: $f(x) = \frac{3x - 1}{2x - 8}$
Find the domain of the following rational function:
$f(x) = \frac{3x - 1}{2x - 8}$
Find the domain of the following rational function with a quadratic denominator: $f(x) = \frac{x}{x^2 - 9}$
Find the domain of the following rational function with a quadratic denominator:
$f(x) = \frac{x}{x^2 - 9}$
Functions with roots
Calculate the domain of definition of the radical function: $f(x) = \sqrt{3x + 12}$
Calculate the domain of definition of the radical function:
$f(x) = \sqrt{3x + 12}$
Calculate the domain of the following function containing a radical with an odd index: $f(x) = \sqrt[3]{2x - 7}$
Calculate the domain of the following function containing a radical with an odd index:
$f(x) = \sqrt[3]{2x - 7}$
Determine the domain of the following function with a square root: $f(x) = \sqrt{x^2 - 5x + 6}$
Determine the domain of the following function with a square root:
$f(x) = \sqrt{x^2 - 5x + 6}$
Logarithmic and combined functions
When a function combines several restrictions (a root inside a fraction, for example), you work out all the conditions separately, and the final domain is where all of them hold at the same time.
Determine the domain of the following logarithmic function: $f(x) = \log_2(5 - x)$
Determine the domain of the following logarithmic function:
$f(x) = \log_2(5 - x)$
Find the domain of definition of the function that combines a fraction and a root: $f(x) = \frac{5x}{\sqrt{2x + 6}}$
Find the domain of definition of the function that combines a fraction and a root:
$f(x) = \frac{5x}{\sqrt{2x + 6}}$
Calculate the domain of the following logarithmic function with a quadratic argument: $f(x) = \ln(9 - x^2)$
Calculate the domain of the following logarithmic function with a quadratic argument:
$f(x) = \ln(9 - x^2)$
Determine the domain of the function that includes a root in the numerator and a polynomial in the denominator: $f(x) = \frac{\sqrt{x + 4}}{x - 2}$
Determine the domain of the function that includes a root in the numerator and a polynomial in the denominator:
$f(x) = \frac{\sqrt{x + 4}}{x - 2}$
Domain and range from the graph
Vertical asymptotes (excluded points of the domain where the graph "shoots off" upwards or downwards) and open/closed endpoints must be marked carefully: a filled circle means that value does belong to the function; an open one means it does not.
From the following graphical representation of the function $f(x)$, determine its domain and range.
From the following graphical representation of the function $f(x)$, determine its domain and range.
Observe the graph of the piecewise function $g(x)$. Determine its domain and range, paying special attention to the vertical asymptote and to the points where the…
Observe the graph of the piecewise function $g(x)$. Determine its domain and range, paying special attention to the vertical asymptote and to the points where the intervals are open or closed.
Domain in problems with a context
In geometry or economics problems, the "mathematical" domain of the formula always has an extra restriction: the quantities (lengths, times, amounts) must make physical sense.
A car park charges $2$ € for each hour or fraction of an hour that a vehicle remains parked, with a maximum limit of $12$ hours per day. Write the function relating the…
A car park charges $2$ € for each hour or fraction of an hour that a vehicle remains parked, with a maximum limit of $12$ hours per day. Write the function relating the parking time $x$ (in hours) to the amount to be paid $f(x)$, and determine its domain and range in the context of this problem.
A small toy rocket is launched vertically upwards from the ground. The height $h$ (in metres) it reaches as a function of time $t$ (in seconds) is given by the…
A small toy rocket is launched vertically upwards from the ground. The height $h$ (in metres) it reaches as a function of time $t$ (in seconds) is given by the expression $h(t) = 20t - 5t^2$. Determine the domain and range of this function, knowing that the motion ends when the rocket hits the ground again.
A company that manufactures electronic components has determined that the daily revenue from selling $x$ components is given by $I(x) = 150x - x^2$, and the total…
A company that manufactures electronic components has determined that the daily revenue from selling $x$ components is given by $I(x) = 150x - x^2$, and the total production costs by $C(x) = 2000 + 10x$. Knowing that the factory has a maximum production capacity of $100$ components per day, determine the profit function $B(x)$ and its valid domain in this economic context.