2.3: Using the Graph
The graph tells us a wealth of information about a function.
Domain
Finding domain is as simple as dropping points down to the $x$-axis.
Domain from a Graph
Imagine the points drop straight down by gravity:
Determine the domain of the function $f(x) = \sqrt{4 - x^2}$.
Solving Equations and Inequalities
Solving $f(x) = g(x)$ and $f(x) < g(x)$
The solutions to the equation $f(x) = g(x)$ are the $x$-values where the graphs intersect.
The solutions to the inequality $f(x) < g(x)$ are the $x$-values where the graph of $f(x)$ is under $g(x)$.
The solutions to the equation $f(x) = g(x)$ are the $x$-values where the graphs intersect.
The solutions to the inequality $f(x) < g(x)$ are the $x$-values where the graph of $f(x)$ is under $g(x)$.
Solve the following equation and inequality:
- $2x^2 + 3 = 5x + 6$
- $2x^2 + 3 \leq 5x + 6$
Solve the inequality\[$x^3 + 6 \geq 2x^2 + 5x\]
Hint Subtract both sides by $2x^2 + 5x$ and solve $x^3 - 2x^2 - 5x + 6 \geq 0$ instead.
Increasing/Decreasing Functions
increasing, decreasing function
$f$ is increasing on an interval $I$ if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in $I$.
$f$ is decreasing on an interval $I$ if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ in $I$.
$f$ is increasing on an interval $I$ if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in $I$.
$f$ is decreasing on an interval $I$ if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ in $I$.
Find the intervals where \[f(x) = 12x^2 + 4x^3 - 3x^4\] are increasing and decreasing.
Find the intervals where \[f(x) = x^{2/3}\] are increasing and decreasing.
Local Maxima and Minima
local maximum, local minimum
The number $f(a)$ is a local maximum value of $f$ if $f(a) \geq f(x)$ when $x$ is near $a$.
The number $f(a)$ is a local minimum value of $f$ if $f(a) \leq f(x)$ when $x$ is near $a$.
The number $f(a)$ is a local maximum value of $f$ if $f(a) \geq f(x)$ when $x$ is near $a$.
The number $f(a)$ is a local minimum value of $f$ if $f(a) \leq f(x)$ when $x$ is near $a$.
Find all local minima and maxima for the function \[f(x) = 12x^2 + 4x^3 - 3x^4\]