Pre-Calculus: Understanding Functions

Introduction

• Key topic: Functions
  • How to graph functions
  • Domain and range of functions
• Overview of parent functions and their characteristics

Basic Parent Functions

Linear Function: $y = x$

• Graph: Straight line through the origin
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

Quadratic Function: $y = x^2$

• Graph: Parabola opening upwards
• Domain: $(-\infty, \infty)$
• Range: $[0, \infty)$

Cubic Function: $y = x^3$

• Graph: S-shaped curve
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

Square Root Function: $y = \sqrt{x}$

• Graph: Curve starting at the origin
• Domain: $[0, \infty)$
• Range: $[0, \infty)$

Cube Root Function: $y = \sqrt[3]{x}$

• Graph: S-shaped curve through the origin
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

Absolute Value Function: $y = |x|$

• Graph: V-shaped graph
• Domain: $(-\infty, \infty)$
• Range: $[0, \infty)$

Rational Function: $y = \frac{1}{x}$

• Graph: Hyperbola with asymptotes
• Domain: $(-\infty, 0) \cup (0, \infty)$
• Range: $(-\infty, 0) \cup (0, \infty)$

Exponential Function: $y = e^x$

• Graph: Exponential growth
• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$

Logarithmic Function: $y = \ln(x)$

• Graph: Logarithmic growth
• Domain: $(0, \infty)$
• Range: $(-\infty, \infty)$

Trigonometric Functions

Sine Function

• Graph: Sinusoidal wave
• Domain: $(-\infty, \infty)$
• Range: $[-1, 1]$

Cosine Function

• Graph: Sinusoidal wave, starting at 1
• Domain: $(-\infty, \infty)$
• Range: $[-1, 1]$

Tangent Function

• Graph: Repeated pattern with vertical asymptotes
• Domain: $x \neq \frac{\pi}{2} + n\pi$
• Range: $(-\infty, \infty)$

Transformations of Functions

• Vertical Stretch/Compression: Multiplier outside function
• Horizontal Stretch/Compression: Multiplier inside function
• Translations:
  • Horizontal: $f(x - c)$ shifts right
  • Vertical: $f(x) + c$ shifts up
• Reflections:
  • Over x-axis: $-f(x)$
  • Over y-axis: $f(-x)$

Composite Functions

• Definition: $f(g(x))$ - function inside a function
• Example Calculation: Substituting one function into another

Inverse Functions

• Finding Inverse: Swap x and y, solve for y
• Verification: $f(g(x)) = x$ and $g(f(x)) = x$
• Graph: Reflects over the line $y = x$

Conclusion

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• Topics include calculus, algebra, chemistry, and physics.