Problem 30
Question
Form the composition \(f \circ g\) and give the domain. $$f(x)=\sqrt{1-x}, \quad g(x)=2 \cos x \text { for } x \in[0,2 \pi]$$
Step-by-Step Solution
Verified Answer
The composite function \(f \circ g\) is \(\sqrt{1-2 \cos x}\). And the domain of this function is any \(x\) such that \(-1 \leq 2 \cos x \leq 1\).
1Step 1: Understand the functions
Given two functions \(f(x)=\sqrt{1-x}\) and \(g(x)=2 \cos x \) where \(x \in[0,2 \pi]\). The composite function \(f \circ g\) meaning \(f(g(x))\) implies that function \(f\) operates on the result of function \(g\).
2Step 2: Form the composite function
Form the composite function \(f \circ g\) by replacing \(x\) in \(f(x)\) with \(g(x)\). Thus, \(f(g(x)) = \sqrt{1-2 cos x}\).
3Step 3: Determine the domain of composite function
As per the composite function, \(f(g(x)) = \sqrt{1-2 cos x}\) is defined only when \(1-2\ cos x >= 0\), that is the quantity within the square root should be greater than or equal to zero. Also, for \(g(x) = 2 \cos x\), the maximum value that \(g(x)\) takes within the given interval \(x \in[0,2 \pi]\) of domain is \(g(x) = 2\). Hence, the domain of the composite function is for any \(x\) such that \(-1 \leq 2 \cos x \leq 1\).
Key Concepts
Domain of a FunctionTrigonometric FunctionsSquare Root Function
Domain of a Function
When we talk about the domain of a function, we refer to the set of inputs for which the function is defined and provides a valid output.
In the context of composite functions, determining the domain can be a bit trickier. You need to consider not only the restrictions of each individual function but also the combined effect when you "chain" those functions together.
- Start by finding where each function is defined on its own. For example: - If a function involves a square root, like in this case with the function \(f(x) = \sqrt{1-x}\), then the expression under the square root must be non-negative. This means \(1-x \geq 0\) must hold. - For a trigonometric function like \(g(x) = 2 \cos x\), you're often bound by its natural periodicity and range.- Consider the range of the inner function, as it must fall within the domain of the outer function for the composite to be defined. - Substituting \(g(x)\) in \(f(x)\), we have \(f(g(x)) = \sqrt{1 - 2 \cos x}\). For this expression, the inequality \(1 - 2 \cos x \geq 0\) further restricts the domain.Understanding these intersections and overlaps is key in finding the domain of a composite function.
In the context of composite functions, determining the domain can be a bit trickier. You need to consider not only the restrictions of each individual function but also the combined effect when you "chain" those functions together.
- Start by finding where each function is defined on its own. For example: - If a function involves a square root, like in this case with the function \(f(x) = \sqrt{1-x}\), then the expression under the square root must be non-negative. This means \(1-x \geq 0\) must hold. - For a trigonometric function like \(g(x) = 2 \cos x\), you're often bound by its natural periodicity and range.- Consider the range of the inner function, as it must fall within the domain of the outer function for the composite to be defined. - Substituting \(g(x)\) in \(f(x)\), we have \(f(g(x)) = \sqrt{1 - 2 \cos x}\). For this expression, the inequality \(1 - 2 \cos x \geq 0\) further restricts the domain.Understanding these intersections and overlaps is key in finding the domain of a composite function.
Trigonometric Functions
Trigonometric functions are mathematical functions that relate angles of triangles to their sides. They play a crucial role in various fields, from engineering to physics, and are the foundation of periodic phenomena modeling.
- The six main trigonometric functions include: sine, cosine, tangent, cosecant, secant, and cotangent. In this exercise, we focus on the cosine function.- The cosine function \( \cos x \) returns a value between -1 and 1 for any angle \( x \), and it is periodic with a period of \( 2\pi \). This means after every \( 2\pi \) interval, the cosine function repeats its values.- When scaled, such as in \( g(x) = 2 \cos x \), the range becomes \([-2, 2]\), effectively scaling the original range by a factor of 2.
In our context, understanding the range of \( g(x) \) helps determine when the composite function \( f(g(x)) \) is defined. By ensuring \(1 - 2 \cos x\) remains non-negative, we limit \(x\) to values where the output remains valid.
- The six main trigonometric functions include: sine, cosine, tangent, cosecant, secant, and cotangent. In this exercise, we focus on the cosine function.- The cosine function \( \cos x \) returns a value between -1 and 1 for any angle \( x \), and it is periodic with a period of \( 2\pi \). This means after every \( 2\pi \) interval, the cosine function repeats its values.- When scaled, such as in \( g(x) = 2 \cos x \), the range becomes \([-2, 2]\), effectively scaling the original range by a factor of 2.
In our context, understanding the range of \( g(x) \) helps determine when the composite function \( f(g(x)) \) is defined. By ensuring \(1 - 2 \cos x\) remains non-negative, we limit \(x\) to values where the output remains valid.
Square Root Function
A square root function involves finding a number that, when multiplied by itself, gives the original number. In mathematical terms, it's the inverse of squaring a number. The square root function is commonly represented by \( f(x) = \sqrt{x} \).
- The defining attribute of the square root function is that it only produces real numbers when its input is non-negative. This means \( x \geq 0\).- For composite functions involving square roots, like \( f \circ g \), care must be taken to ensure that the argument of the square root remains non-negative. This condition sets restrictions on the domain.- As in our example with \( f(g(x)) = \sqrt{1 - 2 \cos x}\), the square root imposes the condition \(1 - 2 \cos x \geq 0\).
Given its characteristic, the square root functions are ideal for modeling scenarios where negative outputs are not possible, ensuring outputs are always non-negative. It's crucial in analysis to unearth the intervals where such conditions hold true to correctly identify domains.
- The defining attribute of the square root function is that it only produces real numbers when its input is non-negative. This means \( x \geq 0\).- For composite functions involving square roots, like \( f \circ g \), care must be taken to ensure that the argument of the square root remains non-negative. This condition sets restrictions on the domain.- As in our example with \( f(g(x)) = \sqrt{1 - 2 \cos x}\), the square root imposes the condition \(1 - 2 \cos x \geq 0\).
Given its characteristic, the square root functions are ideal for modeling scenarios where negative outputs are not possible, ensuring outputs are always non-negative. It's crucial in analysis to unearth the intervals where such conditions hold true to correctly identify domains.
Other exercises in this chapter
Problem 29
Indicate on a number line the numbers \(x\) that satisfy the condition. \(|x| \leq 0\).
View solution Problem 29
Solve the inequality and express the solution set as an interval or as the union of intervals. $$0
View solution Problem 30
Find the point where the lines intersect. $$l_{1}: 4 x-y=-2, \quad l_{2}: 3 x+2 y-0$$
View solution Problem 30
Give the domain and range of the function. $$g(x)=\frac{1}{\sqrt{4-x^{2}}}$$
View solution