Question

Identify the function family and describe the domain and range. Use a graphing calculator to verify your answer. \(f(x)=-4 x+11\)

Short answer

The function \(f(x)=-4 x+11\) is a linear function. Its domain is all real numbers (\(-\infty < x < \infty\)), and its range is also all real numbers (\(-\infty < f(x) < \infty\)).

Step-by-step solution

Identifying the function family

The function \(f(x)=-4 x+11\) is a linear function. This is determined by its form, which is \(y=mx+b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.

Describing the domain

The domain of a linear function is normally all real numbers, which means it can accept any real number for \(x\). This applies to the function \(f(x) = -4x + 11\), so its domain is \(-\infty < x < \infty\).

Describing the range

The range of a linear function is also typically all real numbers, with the output \(y\) covering all real numbers. The range for the function \(f(x) = -4x + 11\) is therefore \(-\infty < f(x) < \infty\).

Key concepts

Function Family

Understanding the concept of a function family is crucial in mathematics. Essentially, function families are groups of functions that share common characteristics. For the given exercise, the function f(x) = -4x + 11 belong to the linear function family. Linear functions are easily recognizable by their straight-line graphs and standard form y = mx + b, where m represents the slope and b indicates the y-intercept of the line. In this case, the slope (m) is -4 and the y-intercept (b) is 11. Knowing the family a function belongs to can provide immediate information about its behavior and the general shape of its graph.

Linear functions are fundamental in algebra, often serving as a starting point for understanding more complex function families. As one of the simplest forms of functions, their predictability and uniform rate of change make them a key subject for study in introductory mathematics courses.

Domain and Range of a Function

The domain and range of a function are concepts that describe the set of possible inputs and outputs of the function, respectively. For a linear function like f(x) = -4x + 11, the domain is all real numbers because you can substitute any real number for x and the function will still provide an output. Mathematically, this is expressed as (-fty < x < fty). Understanding domain helps students to determine the scope of functional values under consideration.

The range, on the other hand, refers to all possible output values the function can produce, which for linear functions without restrictions is also all real numbers, expressed as (-fty < f(x) < fty). Essentially, this states that as x takes on every possible value, the output f(x) does too. Understanding the range is essential when analyzing the function's output behavior over its entire domain.

Graphing Calculators

Graphing calculators are powerful tools that help students visualize concepts like the domain and range of functions. They can graph a wide variety of functions, which facilitates better comprehension of algebraic relationships. In our exercise, using a graphing calculator to plot the function f(x) = -4x + 11 verifies our understanding of the function's domain and range. By inputting the function into the calculator and viewing the graph, one can see the line extending infinitely in both the positive and negative directions along the x-axis (representing the domain) and y-axis (representing the range).

These devices can also be used to experiment with alterations to the function's formula, such as changes to the slope or y-intercept, giving immediate visual feedback on how these changes affect the graph. This makes graphing calculators invaluable for teaching and learning mathematics, as they provide a hands-on approach to exploring and confirming theoretical concepts.