Suppose That The Function H Is Defined As Follows

The realm of function definition often presents seemingly straightforward equations that, upon closer inspection, reveal layers of complexity and nuance. When presented with a function, such as the hypothetical function h, it's crucial to dissect its definition, understand its domain and range, and explore its behavior across different inputs. Let's delve into the intricacies of such a function, examining the implications of its specific construction and how it interacts with fundamental mathematical principles.

Defining the Function h

Suppose the function h is defined as follows:

h(x) =

x^2 + 3, if x < 0
5x - 2, if 0 ≤ x ≤ 5
26, if x > 5

This piecewise function presents us with different rules governing the output of h(x), depending on the value of x. It's essential to recognize that each condition applies to a specific interval of the real number line. Let's break down each piece:

For any x less than 0, the function h(x) behaves as a quadratic function, specifically x^2 + 3.
For x values between 0 and 5, inclusive, h(x) is defined as a linear function, 5x - 2.
Finally, for all x greater than 5, the function h(x) simply outputs a constant value of 26.

Domain and Range

Understanding the domain and range of a function is paramount to grasping its complete behavior. The domain represents all possible input values (x values) for which the function is defined. The range represents all possible output values (h(x) values) that the function can produce.

Domain of h

In the case of our function h, the domain is all real numbers. This is because each piece of the function covers a specific interval, and these intervals collectively span the entire real number line. There are no restrictions on the values of x that can be inputted into the function.

Range of h

Determining the range requires a more nuanced approach, considering the behavior of each piece of the function:

For x < 0: h(x) = x^2 + 3 As x approaches negative infinity, x^2 approaches positive infinity. Since we are adding 3 to x^2, the range of this piece extends from 3 (exclusive) to positive infinity. Note that it is exclusive because x is strictly less than 0, so it never reaches 0. As x approaches 0, h(x) approaches 3.
For 0 ≤ x ≤ 5: h(x) = 5x - 2 This is a linear function. When x = 0, h(x) = -2. When x = 5, h(x) = 23. Since the linear function is continuous, this piece covers all values between -2 and 23, inclusive.
For x > 5: h(x) = 26 This piece simply outputs the constant value 26.

Combining these observations, we can determine the overall range of h. The second piece of the function produces values from -2 to 23, inclusive. The first piece produces values greater than 3. The third piece is simply 26. Since the interval [-2, 23] covers all values from -2 to 23, and we have 26, the range then becomes [-2, 23] ∪ {26} ∪ (3, ∞). Since 26 > 23, and (3, ∞) includes all numbers greater than 3, the range simplifies to [-2, ∞).

Evaluating the Function

To solidify our understanding, let's evaluate the function h at various points:

h(-2): Since -2 < 0, we use the first piece: h(-2) = (-2)^2 + 3 = 4 + 3 = 7.
h(0): Since 0 ≤ 0 ≤ 5, we use the second piece: h(0) = 5(0) - 2 = -2.
h(3): Since 0 ≤ 3 ≤ 5, we use the second piece: h(3) = 5(3) - 2 = 15 - 2 = 13.
h(5): Since 0 ≤ 5 ≤ 5, we use the second piece: h(5) = 5(5) - 2 = 25 - 2 = 23.
h(7): Since 7 > 5, we use the third piece: h(7) = 26.

These examples illustrate how the appropriate piece of the function is selected based on the input value x.

Graphing the Function

Visualizing the function through a graph provides further insight into its behavior. The graph of h(x) will consist of three distinct segments:

For x < 0: A portion of the parabola y = x^2 + 3. This will be a section of the parabola opening upwards, with its vertex at (0, 3), but only for x values less than 0.
For 0 ≤ x ≤ 5: A line segment of the line y = 5x - 2. This will be a line with a slope of 5 and a y-intercept of -2, but only for x values between 0 and 5, inclusive.
For x > 5: A horizontal line at y = 26. This will be a straight horizontal line extending to the right for all x values greater than 5.

The graph will have a discontinuity at x = 5, where the linear segment ends at y = 23, and the horizontal line begins at y = 26. This jump in the graph indicates that the function is not continuous at x = 5.

Continuity and Differentiability

Continuity

A function is continuous at a point if there are no breaks or jumps in its graph at that point. For a piecewise function to be continuous at a boundary point (where the definition changes), the values of the adjacent pieces must be equal at that point.

In our case, we need to check continuity at x = 0 and x = 5:

At x = 0:
- The first piece approaches x^2 + 3 = 0^2 + 3 = 3.
- The second piece evaluates to 5x - 2 = 5(0) - 2 = -2. Since 3 ≠ -2, the function is not continuous at x = 0.
At x = 5:
- The second piece evaluates to 5x - 2 = 5(5) - 2 = 23.
- The third piece evaluates to 26. Since 23 ≠ 26, the function is not continuous at x = 5.

Therefore, the function h(x) is discontinuous at both x = 0 and x = 5.

Differentiability

A function is differentiable at a point if its derivative exists at that point. For a piecewise function to be differentiable at a boundary point, it must first be continuous at that point. Furthermore, the derivatives of the adjacent pieces must also be equal at that point.

Since h(x) is not continuous at x = 0 and x = 5, it cannot be differentiable at these points. We can also analyze the derivatives of each piece:

For x < 0: h'(x) = 2x
For 0 < x < 5: h'(x) = 5
For x > 5: h'(x) = 0

Even if the function were continuous, we would need to check if the derivatives match at the boundary points. For instance:

At x = 0:
- The derivative of the first piece approaches 2x = 2(0) = 0.
- The derivative of the second piece is 5. Since 0 ≠ 5, the derivatives do not match.
At x = 5:
- The derivative of the second piece is 5.
- The derivative of the third piece is 0. Since 5 ≠ 0, the derivatives do not match.

Therefore, h(x) is not differentiable at x = 0 and x = 5.

Transformations

Understanding transformations of functions allows us to visualize how the graph of a function changes when we apply certain operations. Common transformations include:

Vertical shifts: Adding a constant to the function shifts the graph vertically.
Horizontal shifts: Adding a constant to the input variable x shifts the graph horizontally.
Vertical stretches/compressions: Multiplying the function by a constant stretches or compresses the graph vertically.
Horizontal stretches/compressions: Multiplying the input variable x by a constant stretches or compresses the graph horizontally.
Reflections: Multiplying the function by -1 reflects the graph across the x-axis. Multiplying the input variable x by -1 reflects the graph across the y-axis.

Consider a transformation of our function h(x), such as g(x) = 2h(x - 1) + 3. This transformation involves the following steps:

Horizontal shift: Replace x with (x - 1), shifting the graph 1 unit to the right.
Vertical stretch: Multiply the function by 2, stretching the graph vertically by a factor of 2.
Vertical shift: Add 3 to the function, shifting the graph 3 units upward.

The transformed function g(x) would then be defined as:

g(x) =

2((x - 1)^2 + 3) + 3, if x - 1 < 0 (x < 1)
2(5(x - 1) - 2) + 3, if 0 ≤ x - 1 ≤ 5 (1 ≤ x ≤ 6)
2(26) + 3, if x - 1 > 5 (x > 6)

Simplifying:

g(x) =

2(x^2 - 2x + 1 + 3) + 3 = 2x^2 - 4x + 8 + 3 = 2x^2 - 4x + 11, if x < 1
2(5x - 5 - 2) + 3 = 2(5x - 7) + 3 = 10x - 14 + 3 = 10x - 11, if 1 ≤ x ≤ 6
52 + 3 = 55, if x > 6

Analyzing the transformed function g(x) in this way allows us to understand how the original function h(x) is modified and how its key characteristics change.

Inverse Functions

An inverse function, denoted as h^(-1)(x), "undoes" the operation of the original function h(x). In other words, if h(a) = b, then h^(-1)(b) = a. A function has an inverse if and only if it is one-to-one (i.e., it passes the horizontal line test). Piecewise functions can have inverses, but we need to consider each piece separately.

However, because our function h(x) is not one-to-one over its entire domain (e.g., the quadratic piece fails the horizontal line test), it does not have a single inverse function that applies to all x values. We could, however, find inverses for restricted domains where the function is one-to-one. Let's examine each piece:

For x < 0: h(x) = x^2 + 3

This portion of the function is not one-to-one over the interval x < 0. Therefore, we cannot define a unique inverse function on this interval.
For 0 ≤ x ≤ 5: h(x) = 5x - 2

This portion of the function is linear and one-to-one. To find the inverse, we solve for x in terms of y:

y = 5x - 2 y + 2 = 5x x = (y + 2) / 5

So, the inverse function for this piece is h^(-1)(x) = (x + 2) / 5. However, we need to consider the domain and range of this inverse. The domain of the inverse is the range of the original piece, which is [-2, 23]. Therefore, h^(-1)(x) = (x + 2) / 5 for -2 ≤ x ≤ 23.
For x > 5: h(x) = 26

This piece is a horizontal line and is not one-to-one. Therefore, we cannot define an inverse function for this piece.

In summary, we can only define an inverse function for the linear piece of h(x) over the interval [0, 5]:

h^(-1)(x) = (x + 2) / 5, for -2 ≤ x ≤ 23

Applications and Modeling

Piecewise functions like h(x) are useful in modeling real-world situations where different rules or conditions apply depending on the value of a particular variable. Here are a couple of examples:

Tax brackets: The amount of income tax you pay often depends on your income level. Different tax rates apply to different income brackets, which can be modeled using a piecewise function.
Shipping costs: The cost of shipping a package might depend on its weight. For example, it might cost a flat fee for packages under a certain weight, and then increase linearly for packages above that weight.
Cell phone plans: Many cell phone plans offer a certain amount of data for a fixed price, and then charge extra for data usage above that limit. This can be modeled as a piecewise function.
Amusement park ticket prices: Amusement parks might offer different ticket prices based on age. For example, children under a certain age might get in for free, adults pay one price, and seniors pay a discounted price.

The flexibility of piecewise functions allows them to capture complex relationships that cannot be easily represented by a single equation.

Advantages and Disadvantages

Like all mathematical tools, piecewise functions have their own set of advantages and disadvantages:

Advantages:

Flexibility: Piecewise functions can model complex relationships that cannot be easily represented by a single equation.
Accuracy: They can provide a more accurate representation of real-world phenomena that involve different rules or conditions.
Control: They give you fine-grained control over the behavior of the function in different regions of its domain.

Disadvantages:

Complexity: Piecewise functions can be more complex to analyze and work with than simpler functions.
Discontinuities: They can introduce discontinuities and non-differentiable points, which can make certain mathematical operations more challenging.
Definition: Defining and maintaining a piecewise function can be more cumbersome than defining a single equation.

Despite these disadvantages, piecewise functions are a valuable tool in many areas of mathematics, science, and engineering. Their ability to model complex relationships makes them indispensable in many applications.

Conclusion

The function h(x), as defined, provides a rich context for exploring fundamental concepts in calculus and analysis. Through examining its domain, range, continuity, differentiability, transformations, and potential applications, we gain a deeper understanding of how functions can be constructed and analyzed. Piecewise functions, like h(x), highlight the importance of careful analysis and consideration of different conditions when dealing with mathematical models. The process of dissecting and understanding such functions strengthens our ability to tackle more complex problems in mathematics and its applications.