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Chapter 1: Problem 7

In Exercises 1-9, match each function with its name. \(f(x) = |x|\) (a) squaring function (b) square root function (c) cubic function (d) linear function (e) constant function (f) absolute value function (g) greatest integer function (h) reciprocal function (i) identity function

Short Answer

Expert verified
The answer is (f) absolute value function.

Step by step solution

01

Identify the Given Function

Look at the function provided, \(f(x) = |x|\). The bars around the x represent the absolute value, which means the result is always positive regardless of whether x is negative or positive.
02

Match the Function Format with the Correct Option

Now, look at the provided options. Find the name of a function type that refers to absolute values. In this case, the answer should be (f) absolute value function, as this describes a function where the result is always positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Types
In mathematics, understanding different function types is crucial for solving equations and graphing them accurately. Functions represent relationships between variables, often showing how one value depends on another. They come in various forms, each with its unique properties. Here are some common types of functions you might encounter:
  • Linear Functions: These functions have a constant rate of change and are represented by the equation \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
  • Quadratic Functions: Also known as squaring functions, these include terms like \(x^2\) and have a characteristic parabolic shape.
  • Absolute Value Functions: Defined by the function \(f(x) = |x|\), where the output is always non-negative, reflecting the magnitude of a number without considering its sign.
  • Square Root Functions: These involve the square root of a variable, formulated as \(f(x) = \sqrt{x}\).
  • Other types include: cubic functions, reciprocal functions, constant functions, and more.
Grasping these fundamentals allows students to categorize functions correctly, an essential skill for more advanced topics.
Precalculus Exercises
Precalculus is a preparatory course that bridges the gap between algebra and calculus. It introduces students to essential concepts, such as functions, trigonometry, and analytical geometry. Exercises in precalculus help students develop the problem-solving skills needed in calculus. When tackling a precalculus exercise, it's important to:
  • Identify the function type: Recognizing whether you're dealing with a linear, quadratic, or absolute value function can guide your approach.
  • Use correct formulas: Knowing the standard forms of different functions helps in simplifying and solving problems.
  • Practice graphing: Visualizing functions using graphs can provide insights into their behavior and transformations.
Consistent practice through exercises builds a solid foundation in precalculus, paving the way for success in calculus.
Matching Functions with Their Names
Matching functions with their names involves recognizing the characteristics and behaviors of different functions. This skill is crucial in mathematics, as it aids in understanding how functions operate and what they represent.Here are some tips for effectively matching functions with their appropriate names:
  • Recognize key symbols and terms: For example, the absolute value function is often symbolized by \(|x|\).
  • Understand outcomes: Identify what the function's result will be. For instance, the absolute value function always produces a non-negative result.
  • Relate patterns: Notice patterns in equations, such as quadratics forming parabolas or linear functions creating straight lines.
By honing these skills, students can efficiently match functions with their names, ensuring a deep-rooted understanding of algebraic and precalculus concepts.

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Most popular questions from this chapter

COLLEGE ENROLLMENT The Pennsylvania State University had enrollments of 40,571 students in 2000 and 44,112 students in 2008 at its main campus in University Park, Pennsylvania. (Source: Penn State Fact Book) (a) Assuming the enrollment growth is linear, find a linear model that gives the enrollment in terms of the year \(t\) where \(t=0\) corresponds to 2000. (b) Use your model from part (a) to predict the enrollments in 2010 and 2015. c) What is the slope of your model? Explain its meaning in the context of the situation.

In Exercises 87-92, use the functions given by \(f(x) = \frac{1}{8}x - 3\) and \(g(x) = x^3\) to find the indicated value or function. \(g^{-1} \circ f^{-1}\)

Graph each of the functions with a graphing utility. Determine whether the function is \(even\), \(odd\), or \(neither\). \(f(x) = x^2 - x^4\) \(g(x) = 2x^3 + 1\) \(h(x) = x^5 - 2x^3 + x\) \(j(x) = 2 - x^6 - x^8\) \(k(x) = x^5 - 2x^4 + x - 2\) \(p(x) = x^9 + 3x^5 - x^3 + x\) What do you notice about the equations of functions that are odd? What do you notice about the equations off unctions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation?

In Exercises 7-14, find the inverse function of \(f\) informally. Verify that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). \(f(x) = x - 4\)

The points at which a graph intersects or touches an axis are called the ________ of the graph.
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