H2 Maths Formulas, Techniques & Graphs >> Functions and Graphs >> Functions >>

Find range of a function by GC

Steps

  1. Use the graphing calculator to sketch graph of the function (with domain entered)

  2. The possible values of y is the range of the function

Example

$$ f : x \mapsto 1 + {1 \over x}, x \in \mathbb{R}, x \ne 0 $$

Graphing the function:

Vertical asymptote x = 0 & horizontal asymptote y = 1

Vertical asymptote x = 0 & horizontal asymptote y = 1

From the graph, $y$ can be any real value except $1$.

The range in set-builder notation is $$ R_f = \{ y \in \mathbb{R} \phantom{.} | \phantom{.} y \ne 1\} $$

The range in interval notation is $$ R_f = ( -\infty, 1) \phantom{.} \cup \phantom{.} (1, \infty) $$

An alternative presentation in interval notation is $$ R_f = \mathbb{R} \backslash \{1 \} $$

 

Example (with restricted domain)

$$ g : x \mapsto x^2 - 2x - 2, \phantom{000} x \in \mathbb{R}, x > -1 $$

Graphing in the GC:

Enter the function with the domain specified

Enter the function with the domain specified

To access this screen and use the relevant inequality:1. Press 2nd, followed by math2. Scroll to the CONDITIONS tab

To access this screen and use the relevant inequality:

1. Press 2nd, followed by math

2. Scroll to the CONDITIONS tab

Sketch from GC

Sketch from GC

Actual sketch:

Remember to ‘exclude’ the point (-1, 1)

Remember to ‘exclude’ the point (-1, 1)

From the graph, the smallest value of $y$ is $-3$ (minimum point).

The range in set-builder notation is $$ R_g = \{ y \in \mathbb{R} \phantom{.} | \phantom{.} y \ge -3 \} $$

The range in interval notation is $$ R_g = [-3, \infty) $$