Meditaliano IMAT Prep

Lesson 2: Functions & Logarithms

This lesson provides a deep dive into the crucial mathematical concepts of functions and logarithms. We'll start with the fundamentals of functions (definition, domain, range) and explore specific examples like linear, quadratic, cubic, absolute value, and trigonometric functions in detail. Then, we will study logarithms, their inverse relationship with exponential functions, and their properties to solve complex equations and inequalities. These tools are indispensable for modeling phenomena in science and engineering.

A Deep Dive into Functions

In mathematics, a function is like a machine. You give it an input, and it gives you back exactly one output. We usually write it as $f(x)$, which is read as "f of x".

• $x$ is the input value.
• $f(x)$ (or $y$) is the output value.
• The Domain is the set of all possible input values ($x$).
• The Range is the set of all possible output values ($f(x)$).

The crucial rule of functions is that for any single input from the domain, there must be exactly one output in the range. This is verified graphically by the Vertical Line Test.

Linear Functions: $y = mx + c$

The simplest polynomial function. The constant $m$ represents the slope, or rate of change. The constant $c$ is the y-intercept, the point $(0, c)$.

Quadratic Functions: $y = ax^2 + bx + c$

The graph is a parabola. The sign of $a$ determines its orientation ($a>0$ opens up, $a<0$ opens down). The vertex is at $x = -b/(2a)$. The discriminant, $\Delta = b^2 - 4ac$, tells us the number of x-intercepts (roots).

Cubic Functions: $y = ax^3 + bx^2 + cx + d$

Cubic functions have a characteristic "S" shape. They can have up to two turning points (a local maximum and a local minimum). The end behavior is determined by the sign of $a$: if $a>0$, the graph rises to the right and falls to the left; if $a<0$, it falls to the right and rises to the left.

Absolute Value Functions: $y = |f(x)|$

The graph of $y=|f(x)|$ is obtained by taking the graph of $y=f(x)$ and reflecting any part that is below the x-axis ($y<0$) across the x-axis.

A Deep Dive into Logarithms

A logarithm answers the question: "What exponent do we need to raise a specific base to, to get another number?" It is the inverse operation to exponentiation.

• Common Log: $\log(x)$ is shorthand for $\log_{10}(x)$.
• Natural Log: $\ln(x)$ is shorthand for $\log_e(x)$, where $e \approx 2.718$.

Exponential Functions: The Foundation

Before understanding logarithms, one must understand exponential functions of the form $f(x)=a^x$. The behavior of the graph depends critically on the base $a$.

Exponential and Logarithmic functions are inverses, reflected across y=x.

Key Logarithm Properties

Solving Logarithmic Inequalities

Solving these requires a careful, step-by-step process:

1. Determine the domain: Find all values of $x$ for which the arguments of all logarithms are positive.
2. Combine logarithms: Use log properties to combine terms into a single logarithm on one or both sides.
3. Solve the inequality:
   • If base $a > 1$, keep the inequality sign: $\log_a f(x) > \log_a g(x) \implies f(x) > g(x)$.
   • If base $0 < a < 1$, flip the inequality sign: $\log_a f(x) > \log_a g(x) \implies f(x) < g(x)$.
4. Find the intersection: The final solution is the intersection of the domain from step 1 and the solution from step 3.

Trigonometric Functions

Trigonometric functions (sin, cos, tan) describe the relationship between angles and side lengths in a right-angled triangle.

Mnemonic: SOH CAH TOA

Common Angles

The Unit Circle

The unit circle, a circle with a radius of 1 centered at the origin (0,0), is the foundation for understanding trigonometry. For any angle $\theta$, the point $(x, y)$ where the angle's terminal side intersects the circle gives us the values:

• $\cos(\theta) = x$ (the x-coordinate)
• $\sin(\theta) = y$ (the y-coordinate)
• $\tan(\theta) = \frac{y}{x}$ (the slope of the radius)

The sign (positive or negative) of the functions depends on the quadrant $\theta$ is in (using the "All Students Take Calculus" mnemonic):

• Quadrant I (0 to $\pi/2$): All positive (sin, cos, tan)
• Quadrant II ($\pi/2$ to $\pi$): Sine positive (cos, tan negative)
• Quadrant III ($\pi$ to $3\pi/2$): Tangent positive (sin, cos negative)
• Quadrant IV ($3\pi/2$ to $2\pi$): Cosine positive (sin, tan negative)

Graphs of Trigonometric Functions

These functions are periodic, repeating their values in regular intervals.

$y = \sin(x)$

• Period: $2\pi$
• Amplitude: 1
• Passes through (0, 0)

$y = \cos(x)$

• Period: $2\pi$
• Amplitude: 1
• Passes through (0, 1)

$y = \tan(x)$

• Period: $\pi$
• Amplitude: Undefined
• Asymptotes: at $x = \frac{\pi}{2} + n\pi$

Amplitude and Period Transformations

These basic graphs can be transformed. For $y = A\sin(Bx)$ or $y=A\cos(Bx)$:

• The Amplitude is $|A|$. This stretches the graph vertically.
• The Period is $\frac{2\pi}{|B|}$. This compresses or stretches the graph horizontally.

Inverse Functions

An inverse function, written $f^{-1}(x)$, "undoes" the action of the original function $f(x)$.

The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ across the line $y = x$. For a function to have an inverse, it must be one-to-one (it must pass the Horizontal Line Test).

Practice Problems

Test your understanding. Answer all questions and then click "Submit Answers" to see your score.