Finding the Domain and Range of Several Common Functions

In this lesson, we'll dive into the captivating world of functions, focusing on linear functions, quadratic functions, and square root functions. We'll also unravel the mysteries of inverse functions and explore how their domains and ranges relate to the original functions.

Linear Functions:

Definition: A linear function is an algebraic expression of the form \(f(x)=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept.

Finding Domain and Range:

1. Domain: The domain of a linear function is all real numbers (\(\mathbb{R}\)).
2. Range: The range of a linear function is also all real numbers (\(\mathbb{R}\)). Every possible y-value is achievable as the line extends infinitely in both directions.

Quadratic Functions:

Definition: A quadratic function takes the form \(f(x)=ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants.

Finding Domain and Range:

1. Domain: The domain of a quadratic function is all real numbers (\(\mathbb{R}\)).
2. Range: To find the range, consider the vertex of the parabola. If \(a>0\), the parabola opens upwards, and the range is \([y_{min}​,∞)\). If \(a<0\), the parabola opens downwards, and the range is \((−∞,y_{max}​]\). The minimum or maximum value (\(y_{min}\)​ or \(y_{max}\)​) is found using the formula \(\frac{-\Delta}{4a}\), where \(\Delta\)is the discriminant (\(\Delta = b^2-4ac\)).

Square Root Functions:

Definition: A square root function has the form \(f(x)=ax+b\)​, where \(a\) and \(b\) are constants.

Finding Domain and Range:

1. Domain: For a square root function, the expression inside the square root (\(ax+b\)) must be greater than or equal to zero. So, \(ax+b≥0\) gives the domain.
2. Range: The range is all real numbers greater than or equal to zero because the square root of any real number is non-negative.

Inverse Functions:

Definition: The inverse of a function \(f(x)\) is denoted as \(f^{−1}(x)\) and swaps the roles of \(x\) and \(y\). If \((a,b)\) is on the graph of \(f(x)\), then \((b,a)\) is on the graph of \(f^{−1}(x)\).

Finding Domain and Range:

1. Domain of Inverse: If \(f(x)\) has a domain \(D\), the domain of \(f^{−1}(x)\) is the range of \(f(x)\), and vice versa.
2. Range of Inverse: If \(f(x)\) has a range \(R\), the range of \(f^{−1}(x)\) is the domain of \(f(x)\), and vice versa.

Summary:

• Linear Functions: Infinite domain and range.
• Quadratic Functions: Infinite domain and range determined by the orientation of the parabola.
• Square Root Functions: Domain restricted by the expression inside the square root; range is all non-negative real numbers.
• Inverse Functions: Domains and ranges switch roles between the original function and its inverse.

 Below is a diagram showing some of the functions covered in this lesson:

• In purple is shown an example quadratic function, namely \(y=x^2\)
• In red is shown a linear function \(y=x+1\), and its inverse in blue, \(y=x-1\). Note that the function and its inverse have a line of symmetry along the diagonal \(y=x\). This is the case for all functions and their inverses
• In green is shown a square-root function, \(y=\sqrt{x}\)