Functions, Inverses, and Graphs in Rectangular Coordinates

Functions and Their Inverses

Definition of Inverse Functions

An inverse function reverses the effect of the original function. If $f: X \to Y$ is a function, the inverse function $f^{-1}: Y \to X$ satisfies $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$. The notation $f^{-1}$ is read as "f inverse."

• Key Point: The inverse function is not the same as the multiplicative inverse $\frac{1}{f}$.

• Example: If $f(x) = 2x + 3$, then $f^{-1}(y) = \frac{y - 3}{2}$.

One-to-One Functions

A function has an inverse if and only if it is one-to-one (injective). This means each output is produced by exactly one input.

• Formal Definition: For all $a$ and $b$, if $f(a) = f(b)$ then $a = b$.

• Equivalent Condition: For all $a$ and $b$, if $a \neq b$ then $f(a) \neq f(b)$.

• Example: The function $f(x) = x^2$ is not one-to-one because $f(-1) = f(1) = 1$.

Domain and Range of Inverse Functions

The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

Inverses of Linear Functions

Linear functions of the form $f(x) = ax + b$ (where $a \neq 0$) are always one-to-one and have linear inverses.

• Inverse Formula: If $f(x) = ax + b$, then $f^{-1}(x) = \frac{x - b}{a}$.

• Example: If $f(x) = 3x - 5$, then $f^{-1}(x) = \frac{x + 5}{3}$.

Properties of Inverse Functions

• The composite of two one-to-one functions is also one-to-one.

• $(f^{-1})^{-1} = f$.

• If $f(x) = 0$ has a unique solution, then $x = f^{-1}(0)$.

Restricting Domains to Obtain Inverses

Some functions are not one-to-one on their natural domains but can be made one-to-one by restricting the domain.

• Example: $f(x) = x^2$ is not one-to-one on $(-\infty, \infty)$, but is one-to-one on $[0, \infty)$, where the inverse is $f^{-1}(x) = \sqrt{x}$.

Finding the Inverse of a Function

To find the inverse of a one-to-one function $f$, solve $y = f(x)$ for $x$ in terms of $y$, then replace $y$ with $x$.

• Example: If $f(x) = 2x + 1$, solve $y = 2x + 1$ for $x$: $x = \frac{y - 1}{2}$, so $f^{-1}(x) = \frac{x - 1}{2}$.

Graphing in Rectangular Coordinates

Rectangular Coordinate System

The rectangular coordinate system (or Cartesian plane) is used to graph equations in two variables. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis, intersecting at the origin (0,0).

• Every point in the plane is identified by an ordered pair $(x, y)$.

• Example: The point $P(4, 2)$ is located 4 units right and 2 units up from the origin.

Quadrants of the Coordinate Plane

The plane is divided into four quadrants:

Quadrant	Sign of x	Sign of y
I	+	+
II	-	+
III	-	-
IV	+	-

Graphing Equations

To graph an equation such as $y = x^2 + 2x - 3$, plot points $(x, y)$ that satisfy the equation and connect them smoothly.

• x-intercept: Where the graph crosses the x-axis ($y = 0$).

• y-intercept: Where the graph crosses the y-axis ($x = 0$).

• Example: The graph of $y = x^2 + 2x - 3$ is a parabola.

Intercepts of a Graph

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Key Point: This test ensures each input $x$ has only one output $y$.

Domain and Range from Graphs

The domain of a function is the set of all $x$-values included in the graph, and the range is the set of all $y$-values.

• Example: For $f(x) = \sqrt{x}$, the domain is $[0, \infty)$ and the range is $[0, \infty)$.

Graph of the Absolute-Value Function

The absolute-value function is defined as $f(x) = |x|$.

• Graph: V-shaped, with vertex at the origin.

• Domain: $(-\infty, \infty)$

• Range: $[0, \infty)$

Case-Defined Functions

A case-defined function is defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} x & 0 < x < 3 \\ 4x - 1 & 3 < x < 5 \\ 4 & 5 < x < 7 \end{cases}$

Symmetry in Graphs

Types of Symmetry

• y-axis symmetry: The graph is unchanged when $x$ is replaced by $-x$.

• x-axis symmetry: The graph is unchanged when $y$ is replaced by $-y$.

• Origin symmetry: The graph is unchanged when both $x$ and $y$ are replaced by $-x$ and $-y$.

Testing for Symmetry

Type	Test
y-axis	Replace $x$ by $-x$
x-axis	Replace $y$ by $-y$
Origin	Replace $x$ by $-x$ and $y$ by $-y$

• Example: The graph of $y = x^2$ is symmetric about the y-axis.

Symmetry and Inverse Functions

The graph of a function and its inverse are mirror images across the line $y = x$.

• Example: If $g(x) = 2x + 1$, its inverse $g^{-1}(x) = \frac{x - 1}{2}$ is the reflection of $g(x)$ across $y = x$.

Graphing with Intercepts and Symmetry

Use intercepts and symmetry properties to sketch graphs efficiently.

• If a graph has two types of symmetry (x-axis, y-axis, origin), it must have the third as well.

Summary Table: Quadrants and Symmetry

Quadrant	Coordinates	Sign of x	Sign of y
I	$(x_1, y_1)$	$x_1 > 0$	$y_1 > 0$
II	$(x_2, y_2)$	$x_2 < 0$	$y_2 > 0$
III	$(x_3, y_3)$	$x_3 < 0$	$y_3 < 0$
IV	$(x_4, y_4)$	$x_4 > 0$	$y_4 < 0$

Additional info: These notes cover foundational topics in College Algebra, including functions, inverses, graphing, and symmetry, with examples and definitions suitable for exam preparation.