Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = 3 cos x + 4 sin x (Hint: A trig identity is required.)
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0. Functions
Introduction to Functions
2:43 minutesProblem 1.1.10
Textbook Question
Finding Formulas for Functions
Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.
Verified step by step guidance1
To express the side length of a square as a function of the diagonal length, start by recalling the relationship between the side length (s) and the diagonal (d) of a square. The diagonal divides the square into two right-angled triangles, where the diagonal is the hypotenuse. Using the Pythagorean theorem, we have: \( d = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2} \).
Solve for the side length (s) in terms of the diagonal (d) by rearranging the equation: \( s = \frac{d}{\sqrt{2}} \).
Now, express the area of the square as a function of the diagonal length. The area (A) of a square is given by \( A = s^2 \).
Substitute the expression for s from step 2 into the area formula: \( A = \left(\frac{d}{\sqrt{2}}\right)^2 \).
Simplify the expression for the area: \( A = \frac{d^2}{2} \). This gives the area of the square as a function of the diagonal length.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Relationship Between Side Length and Diagonal of a Square
In a square, the relationship between the side length (s) and the diagonal (d) is defined by the Pythagorean theorem. Specifically, the diagonal can be expressed as d = s√2. This means that to find the side length as a function of the diagonal, we can rearrange this formula to s = d/√2.
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Area of a Square
The area (A) of a square is calculated using the formula A = s², where s is the length of a side. Once we express the side length as a function of the diagonal, we can substitute this expression into the area formula to find the area in terms of the diagonal length.
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Function Notation
Function notation is a way to represent a relationship between variables, typically written as f(x). In this context, we will express the side length and area as functions of the diagonal length, denoting them as s(d) and A(d), respectively. This notation helps clarify how one quantity depends on another.
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