Functions, Transformations, and the Library of Parent Functions

By the end of this lesson, you will be able to identify and graph all parent functions in the standard library, apply the complete set of transformations, and analyze functions' domains, ranges, and key features at a pre-calculus level.

The Library of Parent Functions

Pre-calculus works with a complete library of function types. Mastery of each parent function — its shape, domain, range, and key points — is the foundation for all further analysis.

Function	Domain	Range	Key feature
$f(x) = x$ (linear)	$\mathbb{R}$	$\mathbb{R}$	Slope 1, passes through origin
$f(x) = x^2$ (quadratic)	$\mathbb{R}$	$[0,\infty)$	Parabola, vertex at origin
$f(x) = x^3$ (cubic)	$\mathbb{R}$	$\mathbb{R}$	S-curve through origin
$f(x) = \sqrt{x}$ (square root)	$[0,\infty)$	$[0,\infty)$	Starts at origin, concave down
$f(x) = \sqrt[3]{x}$ (cube root)	$\mathbb{R}$	$\mathbb{R}$	S-curve, inflection at origin
$f(x) =	x	$ (absolute value)	$\mathbb{R}$
$f(x) = 1/x$ (reciprocal)	$\mathbb{R}\setminus{0}$	$\mathbb{R}\setminus{0}$	Hyperbola, asymptotes at $x=0,y=0$
$f(x) = b^x$ (exponential)	$\mathbb{R}$	$(0,\infty)$	Asymptote $y=0$
$f(x) = \log_b x$ (logarithm)	$(0,\infty)$	$\mathbb{R}$	Asymptote $x=0$
$f(x) = \sin x$, $\cos x$ (trig)	$\mathbb{R}$	$[-1,1]$	Periodic, oscillating

Complete Transformation Summary

$$g(x) = a \cdot f\bigl(b(x - h)\bigr) + k$$

Parameter	Description	Effect
$a > 1$	Vertical stretch