Parent Functions Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
Sketch the key features (domain, range, intercepts, asymptotes) of the parent function y=1xy = \frac{1}{x}.

Solution

  1. 1
    Domain: all real numbers except 00, i.e., (−∞,0)∪(0,∞)(-\infty, 0) \cup (0, \infty). Range: same as domain.
  2. 2
    No xx-intercept (numerator is never 00) and no yy-intercept (x=0x = 0 is undefined). Asymptotes: vertical at x=0x = 0, horizontal at y=0y = 0.

Answer

Domain/Range: (−∞,0)∪(0,∞); Asymptotes: x=0,y=0; No intercepts\text{Domain/Range: } (-\infty,0) \cup (0,\infty); \text{ Asymptotes: } x=0, y=0; \text{ No intercepts}
The reciprocal function y=1/xy = 1/x has a hyperbolic shape with two branches in quadrants I and III. It is the parent of the rational function family. All transformations of 1/x1/x preserve the hyperbolic shape but shift, stretch, or reflect the branches.

About Parent Functions

A parent function is the simplest, most basic version of a function family — the unshifted, unstretched, unreflected template. All other functions in the family are transformations of this parent. Memorizing parent function shapes allows rapid graphing of transformed versions.

Learn more about Parent Functions →

More Parent Functions Examples

Example 1 easy

Identify the parent function of [formula] and describe the transformations.

Example 2 medium

Match each equation to its parent function: (a) [formula], (b) [formula], (c) [formula], (d) [formul

Example 4 hard

Write a single equation using transformations of the parent function [formula] that has domain [form

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