Geometry and Functions Exercises
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Questions and Answers

What is the definition of the domain of a function?

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

What is the definition of the range of a function?

The range of a function is the set of all possible output values (y-values) that the function can produce.

What does it mean to determine the sign of a function?

Determining the sign of a function means identifying whether the function's output values are positive, negative, or zero for different intervals of the input.

How do you establish the variation table for a function?

<p>To establish a variation table for a function, you need to find the critical points where the derivative changes sign. These critical points are often the roots of the derivative. Then, by examining the sign of the derivative in each interval determined by the critical points, you can determine whether the function is increasing or decreasing in that interval.</p> Signup and view all the answers

What is the extremum of a function?

<p>The extremum of a function refers to its maximum or minimum values. These points occur where the derivative of the function is equal to zero or undefined.</p> Signup and view all the answers

What does it mean to give the variation table of a function?

<p>Giving the variation table of a function involves presenting the function's behavior in terms of intervals of increasing and decreasing values, based on the critical points where the derivative changes sign.</p> Signup and view all the answers

Explain how to determine the minimum of a function given the expression $f(x) = sin^{2}(x)$

<p>Given the expression $f(x) = sin^{2}(x)$, we can visually determine the minimum by identifying the lowest points on the graph of the function. This would occur at the points where the sine function is equal to -1, which would result in $f(x) = 1$. So the minimum of the function $f(x) = sin^{2}(x)$ is 0.</p> Signup and view all the answers

Flashcards

Domain of Definition

The set of all possible input values (x-values) for which a function is defined.

Antecedent

The input value (x-value) that produces a specific output value (y-value) for a function.

Construct Point M

To find a point M that satisfies the given condition: AM = AB + AC. This involves adding line segments together and finding the resulting point.

Variation of a Function

Describes how the function's output changes as the input changes. This can be shown using a table or a graph.

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Sign of a Function

Indicates whether the output of a function is positive, negative, or zero for different input values.

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Minimum Value of a Function

The smallest output value (y-value) that a function can achieve within a specific interval or its entire domain.

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Solve f(x) = 2

Finding the input values (x-values) that produce a specific output value (y-value) of 2 for the function f(x).

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f(2)

The output value (y-value) of the function f(x) when the input value (x-value) is 2.

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f(-2)

The output value (y-value) of the function f(x) when the input value (x-value) is -2.

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Construct Point N

To find a point N that satisfies the given condition: BN = CA + CB + BC. This involves adding line segments together and finding the resulting point.

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Construct Point P

To find a point P that satisfies the given condition: PA = AC + BC + AB. This involves adding line segments together and finding the resulting point.

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Solve f(x) < -1

Finding the input values (x-values) that produce output values (y-values) less than -1 for the function f(x).

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Parabola

A symmetrical U-shaped curve formed by the intersection of a cone and a plane.

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Interval Notation

A way to represent a set of numbers within a specific range using square brackets and parentheses.

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Df = [-5, 2]

The domain of definition of the function f is from -5 to 2, including both endpoints. This means the function is defined for all x-values between -5 and 2.

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Minimum Value on an Interval

The lowest output value (y-value) that a function reaches within a specific interval.

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Table of Variation

A table that summarizes how the output of a function changes as the input changes. It shows the intervals where the function is increasing, decreasing, or constant.

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Table of Sign

A table that shows whether the output of a function is positive, negative, or zero for different intervals of input values.

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Study Notes

Exercise 01

  • Triangle ABC is given
  • Point M is constructed such that AM = AB + AC
  • Point N is constructed such that BN = BC + CA
  • Point P is constructed such that PA = AB + BC + CA

Exercise 02

  • Function f is defined on the domain [-5, 2]
  • Find the domain of definition of f
  • Calculate f(2) and f(-2)
  • Find the values that give f(x) = -1
  • Solve f(x) < -1
  • Create a table of variations for f
  • Create a table of the sign of f
  • Find the minimum of f
  • Solve f(x) = 2
  • Determine the minimum of f on the interval [-2, 4]

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Description

This quiz covers essential exercises related to triangles and functions. In the first exercise, you will explore properties of triangle ABC, including point constructions based on side lengths. The second exercise focuses on function analysis within a defined interval, including domain, value calculations, and sign variation.

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