10 Questions
If $f(x) = 2\sin{x}$ and $g(x) = \cos{x}$, then what is the value of $f(g(\frac{\pi}{4}))$?
2/\sqrt{2}
If $P(x) = (x + 2)^2$ and $Q(x) = x^2 - 4$, what is the value of $P(Q(2))$?
64
If $f(x) = \cos{(2x)}$ and $g(x) = x^2$, what is the derivative of the composite function $f(g(x))$?
-4x \sin{(2x^2)}
If $f(x) = x^2 + 3$ and $g(x) = x - 2$, what is the value of $(f \circ g)^{-1}(7)$?
5
If $P(x) = x^2 + 2x + 1$ and $Q(x) = x - 1$, what is the value of $P(Q^{-1}(x))$?
x^2 - 2x + 1
What is the value of sin(π/6) + cos(π/3)?
1/2
What is the value of (1 + sin(x))^2 + (1 - sin(x))^2?
2
What is the equation of the axis of symmetry for the quadratic function f(x) = x^2 - 4x + 3?
x = 2
If sin(x) = 2/3, what is the value of cos(x)?
√5/3
What is the value of the expression (cos(x) + sin(x))^2 - (cos(x) - sin(x))^2?
2sin(x)cos(x)
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.
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.
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