Finding The Inverse And Domain/Range Of F(x) = X^2 - 13

Let's explore the function f(x) = x² - 13, where x ≥ 0. We'll tackle finding its inverse, graphing both functions, and determining their domains and ranges. This should provide a solid understanding of inverse functions and their properties.

(a) Finding the Equation for f⁻¹(x)

To find the inverse of the function, f⁻¹(x), we need to switch the roles of x and y in the original equation and then solve for y. Here’s a step-by-step breakdown:

1. Replace f(x) with y: y = x² - 13

2. Swap x and y: x = y² - 13

3. Solve for y: x + 13 = y² y = ±√(x + 13)

4. Consider the Domain of f(x): Since the original function f(x) is defined for x ≥ 0, the range of the inverse function f⁻¹(x) must also be non-negative. Therefore, we only consider the positive square root.

5. Write the Inverse Function: f⁻¹(x) = √(x + 13)

Therefore, the equation for the inverse function is f⁻¹(x) = √(x + 13). This inverse function essentially undoes what the original function does. To really understand this, let's break it down. The original function f(x) takes a non-negative number, squares it, and then subtracts 13. The inverse function f⁻¹(x) takes a number, adds 13 to it, and then takes the square root. Notice how the operations are reversed! The domain restriction on the original function is crucial because it ensures that the inverse is also a function. If we didn't have the restriction x ≥ 0, the original function wouldn't be one-to-one, and its inverse wouldn't be a function. Understanding the concept of inverse functions helps in solving equations, simplifying expressions, and analyzing mathematical models. Remember to always consider the domain restrictions when finding and working with inverse functions. Also, always make sure to verify the inverse by checking if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This verification step confirms that you have correctly found the inverse function.

(b) Graphing f(x) and f⁻¹(x) in the Same Coordinate System

To visualize the relationship between f(x) and f⁻¹(x), we'll graph both functions on the same rectangular coordinate system. The graph of a function and its inverse are reflections of each other across the line y = x. This symmetry is a key characteristic of inverse functions.

1. Graphing f(x) = x² - 13 for x ≥ 0: This is a parabola that opens upwards, with its vertex at (0, -13). However, since x ≥ 0, we only consider the right half of the parabola. Plotting a few points can help sketch the graph accurately. For example:

   • When x = 0, f(x) = -13
   • When x = √13, f(x) = 0
   • When x = 4, f(x) = 3 Connect these points with a smooth curve to represent f(x).

2. Graphing f⁻¹(x) = √(x + 13): This is a square root function that starts at the point (-13, 0) and increases as x increases. Plotting a few points can help:

   • When x = -13, f⁻¹(x) = 0
   • When x = 0, f⁻¹(x) = √13
   • When x = 3, f⁻¹(x) = 4 Connect these points with a smooth curve to represent f⁻¹(x).

3. The Line of Symmetry (y = x): Draw the line y = x on the same coordinate system. This line acts as a mirror. If you were to fold the graph along this line, the graphs of f(x) and f⁻¹(x) would overlap. This visual representation provides an intuitive understanding of how a function and its inverse relate to each other. The graph of f(x) is the right half of a parabola opening upwards with its vertex at (0,-13), and the graph of f⁻¹(x) is a square root function starting at (-13,0) and increasing as x increases. The line y=x serves as the axis of symmetry. To master this concept, try plotting more points for each function and observe how they are reflected across the line y=x. Also, practice with different functions to build your intuition for how inverse functions behave graphically.

The graphical representation solidifies the understanding of inverse functions. It helps visualize the domain and range restrictions and the symmetry between the function and its inverse. When sketching the graphs, remember to pay attention to key points and the overall shape of the functions. By accurately plotting and connecting the points, you can gain a deeper understanding of the relationship between a function and its inverse.

(c) Domain and Range of f(x) and f⁻¹(x) using Interval Notation

Determining the domain and range of a function and its inverse is crucial for understanding their behavior and limitations. The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For inverse functions, the domain of f(x) is the range of f⁻¹(x), and the range of f(x) is the domain of f⁻¹(x).

For f(x) = x² - 13, x ≥ 0:

• Domain: Since x ≥ 0, the domain of f(x) is [0, ∞). This means that x can be any non-negative real number.
• Range: The minimum value of f(x) occurs when x = 0, which gives f(0) = -13. As x increases, f(x) also increases without bound. Therefore, the range of f(x) is [-13, ∞). This signifies that y can be any real number greater than or equal to -13.

For f⁻¹(x) = √(x + 13):

• Domain: The expression inside the square root must be non-negative, so x + 13 ≥ 0, which means x ≥ -13. Therefore, the domain of f⁻¹(x) is [-13, ∞). This indicates that x can be any real number greater than or equal to -13.
• Range: Since the square root function always returns non-negative values, the range of f⁻¹(x) is [0, ∞). This tells us that y can be any non-negative real number.

Summary Table:

This analysis reveals a clear relationship: the domain of one function is the range of the other, and vice versa. Understanding how to determine the domain and range is crucial for working with functions and their inverses. Always consider any restrictions on the input values and how they affect the output values. By carefully analyzing the function, you can accurately determine its domain and range, providing a comprehensive understanding of its behavior. Practice with various functions to hone your skills in identifying domains and ranges, which is fundamental in many areas of mathematics.

In conclusion, we have successfully found the inverse function, graphed both the original function and its inverse, and determined their domains and ranges. This exercise highlights the fundamental concepts of inverse functions and their properties.

For further reading on inverse functions, you can visit Khan Academy's Inverse Functions Page.