How Do Different Types of Functions Affect Their Domain and Range?

Understanding the domain and range of functions is important when studying algebra and precalculus.

• Domain means the set of possible input values, often called $x$.
• Range means the set of possible output values, often called $y$.

Each type of function has unique traits that affect its domain and range. Let’s look at different types of functions and how they change these sets.

1. Linear Functions

Linear functions are written as $y = mx + b$.

• Domain: All real numbers, which we write as $(-\infty, \infty)$.
• Range: Also all real numbers $(-\infty, \infty)$.

This happens because a line goes on forever in both horizontal and vertical directions.

2. Quadratic Functions

Quadratic functions look like this: $y = ax^2 + bx + c$, where $a$ is not zero.

• Domain: All real numbers, $(-\infty, \infty)$.
• Range: This depends on $a$:
  • If $a > 0$, the function opens upwards. The range is $[k, \infty)$, where $k$ is the lowest point (the vertex’s y-value).
  • If $a < 0$, it opens downwards. The range is $(-\infty, k]$.

3. Polynomial Functions

Polynomial functions can be written as $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

• Domain: All real numbers, $(-\infty, \infty)$.
• Range: This changes depending on the highest degree:
  • Odd-degree polynomials have a range of all real numbers $(-\infty, \infty)$.
  • Even-degree polynomials have a range that depends on whether the leading number is positive or negative. It could be $[k, \infty)$ or $(-\infty, k]$.

4. Rational Functions

A rational function looks like this: $f(x) = \frac{p(x)}{q(x)}$, with both $p(x)$ and $q(x)$ being polynomials.

• Domain: We need to avoid any value that makes the bottom part ($q(x)$) equal to zero. For example, if $q(x) = x - 2$, then $x=2$ cannot be in the domain.
• Range: It can be tricky to find the range. We often look at certain lines called asymptotes to help us understand the behavior of the function.

5. Radical Functions

Radical functions include square roots, cube roots, and other root types, usually written as $y = \sqrt[n]{x}$.

• Domain: For square roots (even roots), $x$ must be zero or positive. So, for $y = \sqrt{x}$, the domain is $[0, \infty)$. For odd roots, the domain includes all real numbers $(-\infty, \infty)$.
• Range: This depends on the type of root:
  • For even roots, the range is $[0, \infty)$.
  • For odd roots, the range is $(-\infty, \infty)$.

6. Exponential and Logarithmic Functions

Exponential functions are written as $y = a^x$, where $a$ is positive. Logarithmic functions are the opposite of exponentials.

• Exponential Functions:
  • Domain: $(-\infty, \infty)$.
  • Range: $(0, \infty)$.
• Logarithmic Functions:
  • Domain: $(0, \infty)$ (logs can't work with zero or negative numbers).
  • Range: $(-\infty, \infty)$.

In conclusion, the kind of function we use plays a big role in its domain and range. This helps us understand how the function behaves in different situations.