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Questions and Answers
If $f(x) = 2\sin{x}$ and $g(x) = \cos{x}$, then what is the value of $f(g(\frac{\pi}{4}))$?
If $f(x) = 2\sin{x}$ and $g(x) = \cos{x}$, then what is the value of $f(g(\frac{\pi}{4}))$?
If $P(x) = (x + 2)^2$ and $Q(x) = x^2 - 4$, what is the value of $P(Q(2))$?
If $P(x) = (x + 2)^2$ and $Q(x) = x^2 - 4$, what is the value of $P(Q(2))$?
If $f(x) = \cos{(2x)}$ and $g(x) = x^2$, what is the derivative of the composite function $f(g(x))$?
If $f(x) = \cos{(2x)}$ and $g(x) = x^2$, what is the derivative of the composite function $f(g(x))$?
If $f(x) = x^2 + 3$ and $g(x) = x - 2$, what is the value of $(f \circ g)^{-1}(7)$?
If $f(x) = x^2 + 3$ and $g(x) = x - 2$, what is the value of $(f \circ g)^{-1}(7)$?
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If $P(x) = x^2 + 2x + 1$ and $Q(x) = x - 1$, what is the value of $P(Q^{-1}(x))$?
If $P(x) = x^2 + 2x + 1$ and $Q(x) = x - 1$, what is the value of $P(Q^{-1}(x))$?
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What is the value of sin(π/6) + cos(π/3)?
What is the value of sin(π/6) + cos(π/3)?
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What is the value of (1 + sin(x))^2 + (1 - sin(x))^2?
What is the value of (1 + sin(x))^2 + (1 - sin(x))^2?
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What is the equation of the axis of symmetry for the quadratic function f(x) = x^2 - 4x + 3?
What is the equation of the axis of symmetry for the quadratic function f(x) = x^2 - 4x + 3?
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If sin(x) = 2/3, what is the value of cos(x)?
If sin(x) = 2/3, what is the value of cos(x)?
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What is the value of the expression (cos(x) + sin(x))^2 - (cos(x) - sin(x))^2?
What is the value of the expression (cos(x) + sin(x))^2 - (cos(x) - sin(x))^2?
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Study Notes
Here are the study notes for the given topics:
Trigonometry
- Trigonometry deals with the relationships between the sides and angles of triangles.
- It involves the use of trigonometric functions such as sine, cosine, and tangent to solve problems.
- Trigonometric functions can be used to model periodic phenomena, such as sound waves or light waves.
Rules of Sign and Cosine
- The sign rule states that the sign of a trigonometric function depends on the quadrant in which the angle lies.
- The cosine rule states that the cosine of an angle is positive in the first and fourth quadrants, and negative in the second and third quadrants.
- These rules can be used to determine the sign of a trigonometric function without having to calculate its value.
Binomials
- A binomial is an expression consisting of two terms, such as x + 3 or x^2 - 4.
- Binomials can be expanded using the distributive property, which states that a(b + c) = ab + ac.
- The binomial theorem provides a formula for expanding powers of binomials, such as (x + y)^n.
Quadratic Functions
- A quadratic function is a polynomial function of degree two, with the general form f(x) = ax^2 + bx + c.
- The graph of a quadratic function is a parabola, which opens upward or downward depending on the sign of the coefficient a.
- Quadratic functions can be factored using the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a.
Composite Functions
- A composite function is a function that is formed by combining two or more functions, such as f(g(x)) or (f(x) + g(x)).
- The domain of a composite function is the intersection of the domains of the individual functions.
- Composite functions can be used to model complex relationships between variables.
Inverse Function
- An inverse function is a function that reverses the operation of another function, such as f^-1(x) = y if f(y) = x.
- Inverse functions can be used to solve equations, such as finding the value of x that satisfies f(x) = y.
- The inverse of a function can be found using the algebraic method or the graphical method.
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Description
Test your understanding of trigonometry, algebra, and function concepts, including rules of sign, cosine, binomials, quadratic functions, composite functions, and inverse functions. Choose the correct answers from the options.
