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 Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(I), which requires students to determine and analyze the domain, range, symmetry, extrema, zeros, and asymptotes of a function.
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.