Precalculus Study Guide: Functions, Polynomials, Quadratics, Rational Expressions, Variation, Exponential and Logarithmic Forms

Section 2.6 – Composition of Functions

Understanding Function Composition

Function composition involves applying one function to the results of another. If $f(x)$ and $g(x)$ are functions, the composition $f(g(x))$ means you substitute $g(x)$ into $f(x)$.

• Definition: The composition of $f$ and $g$ is written as $(f \\circ g)(x) = f(g(x))$.

• Order Matters: $f(g(x))$ is generally not the same as $g(f(x))$.

• Example: If $f(x) = x^2 + 2x$ and $g(x) = x - 5$, then $f(g(x)) = (x - 5)^2 + 2(x - 5)$.

Graphical Interpretation: You can use graphs of $f$ and $g$ to evaluate $f(g(x))$ for specific values.

Section 2.7 – Inverse Functions

Finding and Understanding Inverses

An inverse function reverses the effect of the original function. If $f(x)$ maps $x$ to $y$, then $f^{-1}(y)$ maps $y$ back to $x$.

• Definition: $f^{-1}(x)$ is the inverse of $f(x)$ if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Finding Inverses: To find the inverse, solve $y = f(x)$ for $x$ and then swap $x$ and $y$.

• Example: For $f(x) = 3x - 7$, set $y = 3x - 7$, solve for $x$: $x = rac{y + 7}{3}$, so $f^{-1}(x) = rac{x + 7}{3}$.

• Domain and Range: The domain of $f$ becomes the range of $f^{-1}$ and vice versa.

Section 3.1 – Quadratic Applications

Modeling and Solving Quadratic Problems

Quadratic functions are used to model various real-world scenarios, such as revenue, projectile motion, and optimization problems.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: The vertex $(h, k)$ of $f(x)$ is at $h = -\frac{b}{2a}$, $k = f(h)$.

• Maximum/Minimum: The vertex gives the maximum or minimum value, depending on the sign of $a$.

• Example: For $R(x) = -2x^2 + 40x - 500$, the maximum revenue occurs at $x = -\frac{40}{-4} = 10$.

• Graphing: Identify the axis of symmetry, vertex, and intercepts to sketch the graph.

Section 3.2 – Polynomial Basics

Understanding Polynomials

Polynomials are algebraic expressions consisting of terms with non-negative integer exponents.

• Degree: The highest exponent of $x$ in the polynomial.

• End Behavior: Determined by the leading term; for $ax^n$, as $x \to \infty$, the sign and degree affect the graph's direction.

• Example: $f(x) = -3x^4 + 2x^3(x - 5)$ has degree 4.

• Graphical Analysis: Use the graph to determine properties such as zeros, end behavior, and turning points.

Section 3.3 – Polynomial Division and Roots

Dividing Polynomials and Finding Roots

Polynomial division helps simplify expressions and find roots (solutions where $f(x) = 0$).

• Long Division: Divide polynomials similarly to numbers, subtracting multiples of the divisor.

• Synthetic Division: A shortcut for dividing by linear factors of the form $(x - c)$.

• Finding Roots: After division, set the quotient equal to zero and solve for $x$.

• Example: Divide $2x^3 - 5x^2 + 12x + 4$ by $x^2 + 2x + 3$.

Section 3.5 – Finding the Domain of a Function

Determining Valid Inputs

The domain of a function is the set of all real numbers for which the function is defined.

• Square Roots: The radicand must be non-negative. For $f(x) = \sqrt{7 - x}$, $7 - x \geq 0 \Rightarrow x \leq 7$.

• Rational Functions: The denominator cannot be zero. For $f(x) = \frac{4x}{x^2 - 25}$, $x^2 - 25 \neq 0 \Rightarrow x \neq \pm5$.

• Example: Find the domain of $f(x) = \frac{2x}{2x + 7}$: $2x + 7 \neq 0 \Rightarrow x \neq -\frac{7}{2}$.

Section 3.6 – Rational Inequalities

Solving Inequalities Involving Rational Expressions

Rational inequalities involve expressions with polynomials in the numerator and denominator.

• Steps:

  1. Find critical points by setting numerator and denominator to zero.

  2. Test intervals between critical points.

  3. Write solution set, excluding values that make the denominator zero.

• Example: Solve $\frac{2x - 3}{x + 1} > 0$.

Section 3.7 – Variation

Direct, Inverse, and Joint Variation

Variation describes how one quantity changes in relation to another.

• Direct Variation: $y = kx$; $y$ increases as $x$ increases.

• Inverse Variation: $y = \frac{k}{x}$; $y$ decreases as $x$ increases.

• Joint Variation: $y = kxz$; $y$ varies directly with both $x$ and $z$.

• Example: The intensity of light varies inversely with the square of the distance: $I = \frac{k}{d^2}$.

Section 4.1 – Exponential Growth and Compound Interest

Modeling Growth and Financial Applications

Exponential functions model growth and decay, including compound interest in finance.

• Exponential Growth: $A = P(1 + r)^n$ where $P$ is principal, $r$ is rate, $n$ is periods.

• Compound Interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$ where $n$ is number of compounding periods per year.

• Example: Find the value of $12,500$ invested for $3$years at $4.25\%$ compounded quarterly.

Section 4.2 – Logarithmic and Exponential Forms

Converting Between Logarithmic and Exponential Equations

Logarithms and exponentials are inverse operations. Converting between forms is essential for solving equations.

• Exponential Form: $a^x = b$

• Logarithmic Form: $x = \log_a b$

• Example: $2^x = 7$ can be written as $x = \log_2 7$.

• Example: $\log_x 3 = 5$ can be written as $x^5 = 3$.