Question

if anyone could explain / answer these questions you’d be saving my life fr thank you (will give 15 pts)

Answer 1

The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values that the function can produce for the corresponding input values.

Domain:

- The domain of a function is the set of all possible input values (independent variable) for which the function is defined.

- In non-graphical terms, it represents all the valid inputs that the function can take.

Range:

- The range of a function is the set of all possible output values (dependent variable) that the function can produce for the corresponding input values.

- In non-graphical terms, it represents all the possible values that the function can output.

Non-Graphical Example:

Consider a function that represents the area of a square based on its side length:

`\[ A(\text{side length}) = (\text{side length})^2 \]`

In this case:

- Domain: All real numbers (positive and negative) because you can square any real number.

- Range: All non-negative real numbers because the area of a square is always non-negative.

Connection to Function Families:

Now, let's connect these concepts to the three function families (Linear, Quadratic, Exponential).

1. Linear Functions:

- Domain: All real numbers.

- Range: All real numbers.

- Linear functions have a constant rate of change and form a straight line.

2. Quadratic Functions:

- Domain: All real numbers.

- Range: Depends on the direction of the parabola. If the parabola opens upward, the range is all non-negative real numbers. If it opens downward, the range is all non-positive real numbers.

- Quadratic functions involve squared terms and often represent the shape of a parabola.

3. Exponential Functions:

- Domain: All real numbers.

- Range: All positive real numbers.

- Exponential functions involve a constant base raised to a variable exponent and exhibit rapid growth or decay.

In summary, the domain and range provide information about the possible inputs and outputs of a function. Linear, quadratic, and exponential functions have different characteristics and behaviors, but the concepts of domain and range apply to all of them.

The graphical example of Domain/Range is attached below.