Key features of functions (domain, range, max/min, zeros, asymptotes, intervals)
Key Features of Functions
Key Features of Functions is a topic in Functions in the California Common Core State Standards. It is aligned to Standard F-IF.4, which requires students to interpret the key features of a function from its graph and its equation.
The key features of a function are its domain and range, intercepts and zeros, maxima and minima, intervals of increase and decrease, asymptotes, and symmetry.
Theory
To analyze a function is to describe its most important features. The standard checklist:
- Domain — the allowed inputs; range — the resulting outputs.
- Intercepts / zeros — where the graph meets the axes; zeros are the \(x\)-values with \(f(x)=0\).
- Extrema — local and global maximum and minimum values (turning points).
- Increasing / decreasing intervals — where the graph rises or falls as \(x\) increases.
- Asymptotes — lines the graph approaches without reaching.
- Symmetry — even (\(y\)-axis) or odd (origin).
Two features worth a formula:
For the range, work from the vertex or the end behavior; for domain, exclude inputs that break a square root (negative radicand) or a fraction (zero denominator).
How to analyze a function
- Domain first: exclude zero denominators and negative radicands.
- Intercepts: set \(x=0\) for the \(y\)-intercept; solve \(f(x)=0\) for the zeros.
- Turning points: find maxima/minima (vertex for a parabola).
- Behavior: note increasing/decreasing intervals, asymptotes, and symmetry.
- Range last: read it off once you know the extrema and end behavior.
The radicand must be nonnegative, and a square root returns nonnegative outputs.
| \(x-2\) | \(\ge\) | \(0\Rightarrow x\ge 2\) |
| \(\text{domain}\) | \(=\) | \([2,\infty)\) |
| \(\text{range}\) | \(=\) | \([0,\infty)\) |
Set \(f(x)=0\) and factor.
| \(x^2-x-6\) | \(=\) | \(0\) |
| \((x-3)(x+2)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(3\ \text{ or }\ -2\) |
The zeros are \(x=3\) and \(x=-2\).
The vertex of \(ax^2+bx+c\) is at \(x=-\dfrac{b}{2a}\).
| \(x\) | \(=\) | \(-\dfrac{-4}{2(1)}=2\) |
| \(f(2)\) | \(=\) | \(4-8+1=-3\) |
Vertex \((2,-3)\). The parabola opens up, so \(f\) is decreasing on \((-\infty,2)\) and increasing on \((2,\infty)\).
A value the curve approaches without touching, as \(x\) grows without bound, is a horizontal asymptote.
| \(y\) | \(=\) | \(4\ \text{ (horizontal asymptote)}\) |
It also caps the range from above.
Common pitfalls
Frequently asked questions
What are the key features of a function?
Domain and range, intercepts/zeros, maxima and minima, intervals of increase and decrease, asymptotes, and symmetry.
How do you find the domain and range?
Domain: exclude inputs that make a denominator zero or a radicand negative. Range: read the outputs from the extrema and end behavior.
What is a zero of a function?
An \(x\)-value where \(f(x)=0\) — where the graph crosses or touches the \(x\)-axis. Found by solving \(f(x)=0\).
What does increasing or decreasing mean?
A function is increasing on an interval if its output rises as \(x\) increases there, and decreasing if the output falls.