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Chapter 3: Problem 71

Fill in the blank with the correct term. Some of the given choices will not be used. distance formula, midpoint formula, function, relation, \(x\) -intercept, y-intercept, perpendicular, parallel ,horizontal lines, vertical lines,symmetric with respect to the \(x\) -axis, symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, increasing, decreasing, constant A function \(f\) is said to be ____ on an open interval \(I\) if, for all \(a\) and \(b\) in that interval, \(af(b).

Short Answer

Expert verified
decreasing

Step by step solution

01

- Identify the Interval and the Function's Behavior

Examine the given statement: 'A function f is said to be ____ on an open interval I if, for all a and b in that interval, af(b).' The function behavior relates to how the values of the function change as the variable increases.
02

- Analyze the Implication

Analyze the implication given: 'a < b implies f(a) > f(b).' This describes a situation where as you move to the right (increase x-values) on the interval I, the function values decrease.
03

- Choose the Correct Term

From the list of terms provided, identify the one that describes a function whose values decrease as the input values increase. That term is 'decreasing.'
04

- Verify the Term

Double-check that 'decreasing' fits the given description: A function is decreasing on an interval if for any two points a and b in the interval where a < b, the function value at a is greater than the function value at b.

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Key Concepts

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

Decreasing Functions
In precalculus, understanding the behavior of functions is key. A function is said to be _decreasing_ on an open interval if, for every pair of points within that interval, as the input value increases, the output value decreases. Mathematically, we write this as follows:
  • For all points \(a\) and \(b\) in the interval where \(a < b\), we have \(f(a) > f(b)\).
This means that as you move from left to right along the interval (increasing \(x\)-values), the function values go down. It's the opposite behavior of an increasing function. Decreasing functions can be linear, such as \(f(x) = -2x + 3\), or nonlinear like \(f(x) = -x^2\). Recognizing decreasing functions helps in understanding the overall shape and characteristics of the graph.
Function Behavior
Analyzing how a function behaves on a specific interval is a major part of precalculus. Function behavior refers to descriptions about the function's increase, decrease, or constancy over an interval. To categorize a function's behavior, you should:
  • Determine if as \(x\) moves to the right, \(f(x)\) increases, decreases, or stays constant.
  • Observe critical points where the function changes direction or remains steady.
For example, consider the function \(f(x) = x^3\). As \(x\) increases from left to right, \(f(x)\) increases more rapidly after passing the critical point at \(x = 0\). This tells us that the function is increasing on either side of the critical point. Understanding function behavior helps in sketching graphs and solving real-world problems.
Interval Analysis
Interval analysis involves examining a specific portion of the domain of a function. Open intervals \((a, b)\) are often used to describe where certain behaviors occur. In the context of decreasing functions:
  • Analyze the interval to determine where the function decreases.
  • Verify if the function meets the criteria: for all \(a\) and \(b\) in the interval, \(a < b\) implies \(f(a) > f(b)\).
For instance, analyzing the interval \((1, 3)\) for the function \(f(x) = -x^2\) involves observing that as \(x\) changes from 1 to 3, \(f(x)\) becomes more negative, thus decrease. Interval analysis is critical for understanding where and how functions maintain specific behaviors.
Function Terminology
A clear understanding of function terminology helps in mastering precalculus concepts. Here are some fundamental terms you need to know:
  • Function: A relation where each input (or \(x\)-value) has exactly one output (or \(y\)-value).
  • Interval: A range of values within the domain of a function, can be open \((a, b)\) or closed \([a, b]\).
  • Increasing: A function is increasing on an interval if \(a < b\) implies \(f(a) < f(b)\).
  • Decreasing: As we've discussed, a function is decreasing if \(a < b\) implies \(f(a) > f(b)\).
Utilizing these terminologies will not only help you understand the problems better but also assist in communicating mathematical concepts more clearly.

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