1. Problem statement: You asked for a clear, beginner-friendly, step-by-step guide to Functions and Their Graphs so you can understand from the basics. 2. What is a function: A function is a rule that assigns to each input exactly one output. 3. Notation: We write a function as $f: X \to Y$ and an output as $y=f(x)$ where $x$ is the input and $y$ is the output. 4. Domain and range: The domain is the set of all allowable inputs $x$. 5. The range is the set of all outputs $f(x)$ that actually appear. 6. Example of domain and range: If $f(x)=x^2$ and we consider all real numbers as input then the domain is all real numbers and the range is all real numbers $y\ge 0$. 7. Function types — simple list and example for each: linear: $f(x)=mx+b$. 8. Quadratic: $f(x)=ax^2+bx+c$. 9. Polynomial: sums of powers like $f(x)=x^3-2x+1$. 10. Rational: quotients like $f(x)=\frac{p(x)}{q(x)}$ where $q(x)\neq 0$. 11. Exponential: $f(x)=a^{x}$ with $a>0$, $a\neq 1$. 12. Absolute value: $f(x)=|x|$ which folds negative inputs to positive outputs. 13. Piecewise: different rules on different parts of the domain like $f(x)=\begin{cases}x+1 & x<0 \\ x^2 & x\ge 0\end{cases}$. 14. How to evaluate a function: Substitute the given input into the formula and simplify. 15. Example evaluate: Let $f(x)=2x+3$ and find $f(4)$. 16. Show substitution and simplification: $f(4)=2\cdot 4+3$. 17. Continue simplification: $f(4)=8+3$. 18. Final value: $f(4)=11$. 19. Graphing a function — what the graph shows: The graph plots all pairs $(x,f(x))$ to show input-output pairs. 20. Intercepts: $x$-intercepts are where $f(x)=0$ and $y$-intercept is $f(0)$. 21. Example find intercepts: For $f(x)=x^2-4$ solve $x^2-4=0$. 22. Factor and solve: $x^2-4=(x-2)(x+2)$. 23. So zeros are $x=2$ and $x=-2$ and the $y$-intercept is $f(0)=-4$. 24. Transformations — general rules: For a base function $f(x)$ we get shifts and stretches with the formula $$y=a\,f(x-h)+k$$. 25. Read this rule: $h$ shifts horizontally, $k$ shifts vertically, $a$ stretches or compresses and flips if negative. 26. Example transformation: Start with $f(x)=x^2$ and make $g(x)=2(x-1)^2+3$. 27. Explain step by step: First shift right by 1, then stretch vertically by factor 2, then shift up by 3. 28. Symmetry and parity: A function is even if $f(-x)=f(x)$ which gives symmetry about the $y$-axis. 29. A function is odd if $f(-x)=-f(x)$ which gives rotational symmetry about the origin. 30. Example even/odd: $f(x)=x^2$ is even and $g(x)=x^3$ is odd. 31. Composition of functions: $(f\circ g)(x)=f(g(x))$ means evaluate $g$ first, then $f$. 32. Example composition: If $f(x)=2x+1$ and $g(x)=x^2$ then $(f\circ g)(x)=f(g(x))=2x^2+1$. 33. Inverse functions: The inverse $f^{-1}(x)$ reverses inputs and outputs when possible. 34. How to find an inverse — steps: Replace $f(x)$ with $y$, swap $x$ and $y$, then solve for $y$. 35. Example inverse: If $y=3x+2$ then swap to get $x=3y+2$ and solve for $y$. 36. Show algebra to solve: $x-2=3y$. 37. Divide both sides by 3 and show cancellation explicitly: $$y=\frac{x-2}{3}$$ 38. Another cancellation example when simplifying a rational expression: Start with $$\frac{2x^2+4x}{2x}$$ 39. Factor numerator and show the factored form: $$\frac{2x(x+2)}{2x}$$ 40. Show cancellation using \cancel{...} then simplify: $$\frac{\cancel{2x}(x+2)}{\cancel{2x}}=x+2$$ 41. Note on domains with cancellation: When you cancel factors in a rational expression you must remember the original domain restriction like $x\neq 0$ in this example. 42. Solving equations involving functions: Use algebraic steps and check domain restrictions. 43. Example solve $3x=9$ showing cancellation when dividing: $$3x=9$$ 44. Divide both sides by 3 and show cancellation: $$x=\frac{9}{3}$$ 45. Show cancellation explicitly: $$x=\frac{\cancel{3}3}{\cancel{3}}=3$$ 46. Graphing tips: Make a table of values, plot points, look for symmetry, find intercepts and extrema and use known shapes of basic functions. 47. Extrema and critical points: For calculus you find where slope changes, but for basic graphs look for peaks and valleys visually and via derivatives later. 48. Piecewise graphs: Plot each piece over its domain interval and include closed or open dots depending on whether endpoints are included. 49. Practice problems with short solutions: Example 1 find domain of $f(x)=\frac{1}{x-3}$. 50. Solution: Domain is all real numbers except $x=3$ because denominator cannot be zero. 51. Example 2 find $f^{-1}(x)$ for $f(x)=2x-5$. 52. Solution show steps and cancellation: $y=2x-5$, swap $x=2y-5$, then $x+5=2y$, divide by 2 and show cancellation: $$y=\frac{x+5}{2}$$ 53. Quick reference rules: To shift right by $h$ replace $x$ by $x-h$. 54. To shift up by $k$ add $k$ to the function. 55. To reflect across the $x$-axis multiply the function by $-1$. 56. To reflect across the $y$-axis replace $x$ by $-x$. 57. Final encouragement and next steps: Practice by sketching graphs of $y=x^2$, $y=|x|$, $y=\frac{1}{x}$ and $y=2^x$ and apply transformations one at a time to build intuition. 58. If you want, I can create many short practice problems with step-by-step solutions in this same style.