Different Types of Functions

There are many types of functions, and each one has its own special features. These features affect their domain (the set of possible input values) and range (the set of possible output values). Knowing this is important for studying functions in Algebra I.

1. Linear Functions

• What It Is: A linear function looks like this: ( f(x) = mx + b ). Here, ( m ) is the slope, and ( b ) is where the line crosses the y-axis.
• Domain: You can use any real number as input: all numbers from negative infinity to positive infinity, or ( (-\infty, \infty) ).
• Range: Just like the domain, the range is also all real numbers: ( (-\infty, \infty) ).

2. Quadratic Functions

• What It Is: A quadratic function has the form ( f(x) = ax^2 + bx + c ), with ( a ) not equal to zero.
• Domain: You can also use any real number as input: ( (-\infty, \infty) ).
• Range: The output values depend on the value of ( a ).
  • If ( a ) is positive (greater than zero), the range starts at the minimum point and goes up: ( [k, \infty) ), where ( k ) is the lowest point.
  • If ( a ) is negative (less than zero), the range goes down from the maximum point: ( (-\infty, k] ), where ( k ) is the highest point.

3. Rational Functions

• What It Is: A rational function looks like ( f(x) = \frac{p(x)}{q(x)} ), where ( p ) and ( q ) are polynomial expressions, and ( q(x) ) cannot be zero.
• Domain: The domain will not include any value that makes the bottom (denominator) zero. For example, in ( f(x) = \frac{1}{x-3} ), you can't use ( x = 3 ), so the domain is ( (-\infty, 3) \cup (3, \infty) ).
• Range: The range can be tricky and may involve looking at horizontal asymptotes.

4. Exponential Functions

• What It Is: An exponential function can be shown as ( f(x) = a \cdot b^x ), where both ( a ) and ( b ) are positive numbers.
• Domain: You can use any real number as input: ( (-\infty, \infty) ).
• Range: The output values are greater than zero, so the range is ( (0, \infty) ).

5. Logarithmic Functions

• What It Is: A logarithmic function looks like ( f(x) = \log_b(x) ), where ( b ) is a positive number that is not equal to 1.
• Domain: The input values must be positive: ( (0, \infty) ).
• Range: The range is all real numbers: ( (-\infty, \infty) ).

Conclusion

To sum it up, the type of function you have really affects its domain and range. Linear and quadratic functions can take a wider range of inputs and give a wider range of outputs. On the other hand, rational, exponential, and logarithmic functions have more specific rules for their inputs and outputs due to their unique features. Understanding these types of functions is helpful when analyzing them in algebra.