Problem 58 - Algebra A Combined Function

Question

For each function, find the indicated values. \(h(x)=7\) a. \(h(7)\) b. \(h(542)\) c. \(h\left(-\frac{3}{4}\right)\)

Step-by-Step Solution

Answer

h(x) = 7 for all inputs, so h(7)=7, h(542)=7, h(-3/4)=7.

Step 1: Understanding the Function

The function given is a constant function, \( h(x) = 7 \). In a constant function, the output value (y-value) is the same regardless of the input value (x-value). Therefore, no matter what x is, \( h(x) \) will always be \( 7 \).

Step 2: Evaluate h(7)

Since \( h(x) = 7 \) for all x, the value of \( h(7) \) is 7.

Step 3: Evaluate h(542)

As the function is constant, \( h(542) \) is also equal to 7, because \( h(x) = 7 \) for any value of x.

Step 4: Evaluate h\left(-\frac{3}{4}\right)

Similarly, \( h\left(-\frac{3}{4}\right) = 7 \), since the function is constant and equals 7 for every x value.

Key Concepts

Evaluating Functions

Evaluating functions involves finding the output of a function for a specific input. In simpler terms, it's like asking what the result of a math operation is when using certain numbers. For the function given in the exercise, which is a constant function, evaluating it is quite straightforward.

- A **constant function** is a type of function where the output value is the same for every input.- For instance, if you have a function like \( h(x) = 7 \), this means that no matter what number you plug in for \( x \), the result will always be 7.- Evaluating \( h(7) \), \( h(542) \), or even \( h\left(-\frac{3}{4}\right) \) will all result in 7.

This is because constant functions do not change. Unlike linear or quadratic functions that have variables which influence the output, a constant function remains unchanged regardless of \( x \).

Thus, while evaluating functions of this kind, it becomes more a task of recognizing the function type rather than solving intricate calculations.

Function Notation

Function notation is a way to represent functions in a concise and clear format. In mathematics, it allows us to identify what the function does and how it operates more efficiently.

- Function notation uses symbols like \( f(x) \), \( g(x) \), or \( h(x) \) with \( x \) being the input value and \( f(x) \) being the output.- In our exercise, \( h(x) = 7 \) uses function notation to signify that no matter what \( x \) is, the output is always 7.- This notation is helpful in defining the relationship between elements, indicating what operation is performed, or in the case of a constant function, what the unchanging output is.

Understanding function notation is crucial because it serves as a universal language in algebra and higher mathematics, providing a way to discuss and manipulate functions without ambiguity. It is also essential for communicating mathematical ideas clearly between people.

Functions in Algebra

In algebra, functions are a way to describe the relationship between input and output. They are a core concept, acting as mathematical entities that map values from one set (domain) to another set (range).

- The simplest example is a **constant function**, like \( h(x) = 7 \), where every input has the same output. These help in understanding more complex types of functions.- Functions can be linear, quadratic, polynomial, or exponential, depending on how \( x \) is manipulated.- They let us transform and redefine numbers within equations in a structured way.

Functions play a crucial role in algebra by:

• Enhancing the comprehension of complex relationships between variables.
• Providing tools for modeling real-world phenomena.
• Allowing the solving of equations by replacing variables systematically.

Understanding how functions work aids in building a strong foundation for any further study in mathematics, helping students grasp how variables interact and influence one another.