Question: give the examples of the 7 functions and their concept and also their general formula, constant function, linear function, quadratic function, absolute value function, square root function, piece wise function, and last is rational function. now as i say, give the examples with answer with solutions and explanations with graph and with domain and range
1. **Constant Function**
- Concept: A function that always returns the same value regardless of input.
- General formula: $$f(x) = c$$ where $$c$$ is a constant.
- Example: $$f(x) = 5$$
- Domain: All real numbers $$(-\infty, \infty)$$
- Range: $$\{5\}$$
2. **Linear Function**
- Concept: A function that graphs as a straight line.
- General formula: $$f(x) = mx + b$$ where $$m$$ is slope and $$b$$ is y-intercept.
- Example: $$f(x) = 2x + 3$$
- Domain: $$(-\infty, \infty)$$
- Range: $$(-\infty, \infty)$$
3. **Quadratic Function**
- Concept: A function that graphs as a parabola.
- General formula: $$f(x) = ax^2 + bx + c$$ where $$a \neq 0$$.
- Example: $$f(x) = x^2 - 4x + 3$$
- Domain: $$(-\infty, \infty)$$
- Range: $$[\text{minimum value}, \infty)$$ depending on vertex.
4. **Absolute Value Function**
- Concept: Measures distance from zero, always non-negative.
- General formula: $$f(x) = |x|$$
- Example: $$f(x) = |x - 2|$$
- Domain: $$(-\infty, \infty)$$
- Range: $$[0, \infty)$$
5. **Square Root Function**
- Concept: Outputs the principal square root.
- General formula: $$f(x) = \sqrt{x}$$
- Example: $$f(x) = \sqrt{x - 1}$$
- Domain: $$[1, \infty)$$ (inside root must be $$\geq 0$$)
- Range: $$[0, \infty)$$
6. **Piecewise Function**
- Concept: Different formulas for different parts of domain.
- General formula: $$f(x) = \begin{cases} x+2 & x < 0 \\ x^2 & x \geq 0 \end{cases}$$
- Example: as above
- Domain: $$(-\infty, \infty)$$
- Range: depends on pieces
7. **Rational Function**
- Concept: Ratio of two polynomials.
- General formula: $$f(x) = \frac{p(x)}{q(x)}$$ where $$q(x) \neq 0$$.
- Example: $$f(x) = \frac{1}{x-2}$$
- Domain: $$(-\infty, 2) \cup (2, \infty)$$ (exclude where denominator zero)
- Range: $$(-\infty, 0) \cup (0, \infty)$$
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### Example solution for Linear Function $$f(x) = 2x + 3$$
1. Problem: Find $$f(4)$$.
2. Substitute $$x=4$$: $$f(4) = 2(4) + 3$$.
3. Calculate: $$f(4) = 8 + 3 = 11$$.
4. Domain is all real numbers, range is all real numbers.
### Graphs and domain/range summary:
- Constant: horizontal line at $$y=5$$.
- Linear: straight line with slope 2, intercept 3.
- Quadratic: parabola opening up with vertex at $$x=2$$.
- Absolute value: V shape with vertex at $$x=2$$.
- Square root: curve starting at $$x=1$$ going right.
- Piecewise: line for $$x<0$$, parabola for $$x\geq0$$.
- Rational: hyperbola with vertical asymptote at $$x=2$$.