Precalculus Study Notes: Functions, Inverses, and Exponents

Functions and Their Properties

Function Composition

Function composition involves applying one function to the results of another. If you have two functions, f(x) and g(x), the composition f(g(x)) means you first apply g to x, then apply f to the result.

• Definition: The composition of functions f and g is written as (f ˆ g)(x) = f(g(x)).

• Domain: The domain of f(g(x)) consists of all x in the domain of g such that g(x) is in the domain of f.

• Example: If f(x) = \frac{5}{2x-1} and g(x) = 2x-3, then f(g(x)) = \frac{5}{2(2x-3)-1} = \frac{5}{4x-6-1} = \frac{5}{4x-7}.

Finding the Domain of a Composite Function

To find the domain of a composite function, consider the domains of both functions and any restrictions that arise from the composition.

• Step 1: Find the domain of the inner function g(x).

• Step 2: Find the domain of the outer function f(x) as it applies to g(x).

• Example: For f(g(x)) = \frac{5}{4x-7}, the denominator cannot be zero, so 4x-7 \neq 0 \implies x \neq \frac{7}{4}.

Inverse Functions

Inverse functions "undo" each other. If f(x) and g(x) are inverses, then f(g(x)) = x and g(f(x)) = x for all x in the domain.

• Definition: A function f has an inverse, denoted f^{-1}, if and only if it is one-to-one (1-1).

• One-to-One Function: A function is 1-1 if each output is produced by exactly one input. Formally, f(a) = f(b) \implies a = b.

• Example: If f(x) = \frac{1}{2}x - 5, to find f^{-1}(x):

  • Set y = \frac{1}{2}x - 5

  • Solve for x: y + 5 = \frac{1}{2}x \implies x = 2(y + 5)

  • So, f^{-1}(x) = 2(x + 5)

Testing for Inverses Using Composition

To verify if two functions are inverses, compose them in both orders and check if the result is x:

• f(g(x)) = x and g(f(x)) = x

• Example: If f(x) = \frac{x-1}{x+1} and g(x) = \frac{x+1}{1-x}, compute both compositions to check if they yield x.

Exponents and Simplification

Properties of Exponents

Exponents follow specific rules that allow for simplification:

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

Examples of Exponent Simplification

• Example 1: $16^{1/4} = 2$ because $2^4 = 16$.

• Example 2: $81^{1/2} = 9$ because $9^2 = 81$.

• Example 3: $27^{4/3} = (27^{1/3})^4 = 3^4 = 81$.

• Example 4: $\left(\frac{3}{4xy^2}\right)^{-2} = \left(\frac{4xy^2}{3}\right)^2 = \frac{16x^2y^4}{9}$

• Example 5: $(-3a^2bc^{-1}y^{-2})^2 = 9a^4b^2c^{-2}y^{-4} = \frac{9a^4b^2}{c^2y^4}$

Graphing Linear Functions

Graphing y = 2 + (-2)x

Linear functions are graphed as straight lines. The general form is y = mx + b, where m is the slope and b is the y-intercept.

• Slope (m): The coefficient of x indicates the steepness and direction of the line. Here, m = -2.

• Y-intercept (b): The value where the line crosses the y-axis. Here, b = 2.

• Example: To graph y = 2 - 2x:

  • Start at (0, 2) on the y-axis.

  • From there, move down 2 units and right 1 unit to plot the next point (since the slope is -2).

One-to-One Functions and Inverses

Definition and Importance

A function is one-to-one (1-1) if every output value corresponds to exactly one input value. This property is essential for a function to have an inverse.

• Horizontal Line Test: A function is 1-1 if no horizontal line intersects its graph more than once.

• Why 1-1 is Needed for Inverses: If a function is not 1-1, its inverse would not be a function (it would assign multiple outputs to a single input).

• Example: The function f(x) = x^2 is not 1-1 over all real numbers, but is 1-1 if restricted to x \geq 0.