What is the domain of the real function f(x) = sqrt((3x - 6)/(x + 1))?
Functions
Question · Easy
Given the bijective function f(x) = (2x + 5)/(x - 3) for x != 3, what is its inverse function f^(-1)(x)?
Given the functions f(x) = sqrt(x - 4) and g(x) = x^2 - 5, what is the composite function (f o g)(x) and its natural domain in R?
Question · Medium
Which of the following real functions is an odd function (função ímpar)?
Solve the logarithmic equation ln(x + 2) + ln(x - 1) = ln(10) in the set of real numbers.
What is the range (contradomínio / conjunto de chegada) of the real function f(x) = 4 - 3e^(-2x)?
Determine the value of the constant k such that the piecewise function f(x) is continuous at x = 1, where f(x) = 2x + k for x <= 1 and f(x) = x^2 - 3x + 6 for x > 1.
The graph of y = f(x) has a local maximum at the point P(-2, 5). What are the coordinates of the corresponding local maximum on the graph of g(x) = 3f(x - 4) + 2?
Solve the inequality log_2(x^2 - 3x) <= 2 in the set of real numbers.
Question · Hard
The polynomial P(x) = x^3 - 4x^2 + ax + b leaves a remainder of 2 when divided by (x - 1) and a remainder of -12 when divided by (x + 1). Find the values of a and b.
Calculate the limit: lim_{x -> 4} (sqrt(x) - 2) / (x - 4).
Limits And Continuity
Evaluate the limit at infinity: lim_{x -> +inf} (3x^3 - 5x + 2) / (2x^3 + 7x^2 - 1).
Calculate the fundamental trigonometric limit: lim_{x -> 0} sin(5x) / sin(3x).
Evaluate the limit of difference of radicals: lim_{x -> +inf} (sqrt(x^2 + 6x) - x).
Find the equation of the oblique asymptote (assíntota oblíqua) as x -> +inf for the function f(x) = (2x^2 + 3x - 1)/(x - 1).
Evaluate the notable exponential limit: lim_{x -> +inf} (1 + 3/x)^(2x).
How many total asymptotes (vertical, horizontal, and oblique combined) does the function f(x) = (x^2 - 9)/(x^2 - 4) possess?
For what value of the parameter m is the function f(x) continuous at x = 0, where f(x) = (e^(2x) - 1)/x for x != 0 and f(0) = m?
Calculate the trigonometric limit: lim_{x -> 0} (1 - cos(4x)) / x^2.
According to Bolzano's Theorem (Teorema de Bolzano-Cauchy), on which of the following intervals is the polynomial function P(x) = x^3 - 3x - 5 guaranteed to have at least one real zero?
Given the function f(x) = x^3 e^x, what is its first derivative f'(x)?
Differential Calculus
Calculate the derivative of f(x) = (2x - 1)/(x + 4) evaluated at x = 1.
What is the derivative of the composite trigonometric function f(x) = sin^3(2x)?
Find the equation of the line tangent to the curve y = x^3 - 4x + 3 at the point where x = 2.
On which open interval is the polynomial function f(x) = 2x^3 - 9x^2 + 12x - 5 strictly decreasing?
What is the local maximum value (máximo relativo) of the function f(x) = x^3 - 6x^2 + 9x + 2?
Determine the coordinates of the inflection point (ponto de inflexão) of the curve f(x) = x^3 - 3x^2 - 9x + 4.
Find the derivative of f(x) = ln((x^2 + 1)/(x - 2)) for x > 2.
What is the derivative of the inverse trigonometric function f(x) = arctan(3x)?
Using implicit differentiation, find the slope dy/dx of the curve x^2 + 2xy + y^2 = 9 at the point (1, 2).
Find the equation of the normal line (reta normal) to the curve y = sqrt(2x + 1) at the point where x = 4.
A farmer has 120 meters of fencing to enclose a rectangular plot of land adjacent to an existing straight stone wall (so only three sides require fencing). What is the maximum possible area of the enclosure in square meters?
On which interval is the graph of the function f(x) = x e^(-x) strictly concave downwards (concavidade voltada para baixo)?
An open cylindrical container (without a top lid) has a fixed volume of V = 27pi m^3. What base radius r minimizes the total surface area S of the metal used?
Find the value of c in the open interval (1, 4) that satisfies the Mean Value Theorem (Teorema do Valor Médio de Lagrange) for the function f(x) = x^2 - 2x.
What is the exact value of cos(7pi/6)?
Trigonometry
If sin(theta) = 3/5 and theta in (pi/2, pi), what is the exact value of tan(theta)?
If cos(x) = -1/3, what is the exact value of cos(2x)?
Solve the trigonometric equation 2 sin^2(x) - sin(x) - 1 = 0 in the interval [0, 2pi).
Calculate the exact value of sin(75°).
Simplify the trigonometric expression: (1 - cos(2x)) / sin(2x) for all x where it is defined.
What is the amplitude (amplitude) and fundamental period (período fundamental) of the function f(x) = -4 cos(3x - pi/2) + 1?
Solve the trigonometric inequality 2 cos(x) + 1 < 0 for x in the interval [0, 2pi).
In a triangle ABC, side a = 5, side b = 8, and the included angle C = 60°. What is the exact length of side c?
Calculate the exact value of cos(arcsin(-4/5) + arccos(12/13)).
Find the center C and radius R of the circle with Cartesian equation x^2 + y^2 - 6x + 8y - 11 = 0.
Analytic Geometry
For the ellipse with equation x^2/25 + y^2/9 = 1, what are the coordinates of its foci (focos)?
In an Arithmetic Progression (Progressão Aritmética), the 3rd term is a3 = 11 and the 8th term is a8 = 31. What is the 20th term a20?
Sequences And Series
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