Problem 72 - Calculus

Question

Give an example of: An invertible function whose graph contains the point (0,3).

Step-by-Step Solution

Answer

An example is the function \( f(x) = x + 3 \), which passes through (0,3) and is invertible.

Step 1: Define an Invertible Function

To define an invertible function, we can start with a simple linear function, such as \( f(x) = mx + b \). Linear functions are typically invertible unless they are constant. The function needs to be one-to-one and onto for it to have an inverse.

Step 2: Ensure the Exact Point Fits

To ensure the function passes through the point (0,3), we substitute \((x, y) = (0, 3)\) into \( f(x) \). This gives us \( f(0) = m \times 0 + b = 3 \). So, set \(b = 3\).

Step 3: Choose a Suitable 'M' Value

To ensure the function remains invertible, we choose any non-zero value for \(m\). For simplicity, we choose \(m = 1\), making our function \( f(x) = x + 3 \).

Step 4: Verify Function is Invertible

To confirm \( f(x) = x + 3 \) is invertible, we find its inverse. Solve for \( x \) in terms of \( y \). Starting with \( y = x + 3 \), solve to get \( x = y - 3 \). So, the inverse function is \( f^{-1}(x) = x - 3 \). Both functions are linear and therefore invertible.

Key Concepts

Linear Function

Linear functions are one of the most basic and essential types of functions in mathematics. They have the general form:

• \( f(x) = mx + b \)

where \( m \) and \( b \) are constants.These functions create straight line graphs on the coordinate plane. The constant \( m \) is known as the slope of the line, and it represents the rate of change of the function. The \( b \) constant is the y-intercept, which is the point where the line crosses the y-axis.
Linear functions are very predictable due to their constant rate of change. They are used in many real-world situations where one thing depends linearly on another. As long as \( m eq 0 \), linear functions are invertible, meaning there exists an inverse function for each linear function.

Inverse Function

An inverse function essentially reverses the operation of the original function. For a function to have an inverse, each output should map to exactly one input. If \( f(x) \) takes an input \( x \) and provides an output \( y \), then the inverse function \( f^{-1}(x) \) does the opposite: it takes \( y \) back to \( x \).For a function \( f(x) = x + 3 \), the process of finding an inverse involves swapping the roles of \( x \) and \( y \). Thus:

• Start with \( y = x + 3 \)
• Solve for \( x \), which gives \( x = y - 3 \)
• Thus, the inverse is \( f^{-1}(x) = x - 3 \)

This ability to "reverse" is what makes functions invertible. A linear function will always have an inverse as long as it is not constant.

One-to-One and Onto

For a function to have an inverse, it's essential for the function to be both "one-to-one" and "onto." These two concepts define the nature of the function's invertibility:- **One-to-One:** This means that each input corresponds to a unique output. Mathematically, if \( f(a) = f(b) \) implies that \( a = b \), then the function is one-to-one. In simpler terms, no two different inputs should produce the same output.

- **Onto:** Also called "surjective." This indicates that every possible output in the codomain is mapped to by some input from the domain. Essentially, the function covers all values in the target set. The linear function \( f(x) = x + 3 \) designed in the exercise is both one-to-one and onto, making it invertible. This means every output has an exact pre-image in the domain, fitting all requirements to have an inverse function.