Inverse Functions and Angle Formulas

First, find x in terms of y, then switch x and y.

Because for y = 4, there are two values of x (2 and -2) that satisfy the equation, leading to ambiguity.

The Horizontal Line Test checks if a horizontal line intersects a graph at more than one point, indicating the function has no inverse.

<p>A function has an inverse if it is either strictly increasing or strictly decreasing.</p> Signup and view all the answers

<p>You can determine if f has an inverse by checking if the equation f(x1) = f(x2) implies x1 = x2.</p> Signup and view all the answers

<p>A function must have unique outputs for every input to possess an inverse function.</p> Signup and view all the answers

<p>First, express x in terms of y as x = (y + 2)^(1/3), then switch to get y = (x + 2)^(1/3).</p> Signup and view all the answers

<p>If two different x-values yield the same output, the function does not have an inverse.</p> Signup and view all the answers

<p>cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽.</p> Signup and view all the answers

<p>It expands to 2 - 2 cos(𝛼 − 𝛽), which simplifies using the Pythagorean identity.</p> Signup and view all the answers

<p>cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽.</p> Signup and view all the answers

<p>Combining cos² and sin² terms with Pythagorean identities results in a simplified expression.</p> Signup and view all the answers

<p>The negative sign indicates the sine function of a negative angle, defining its value as a negative.</p> Signup and view all the answers

<p>They equate to 2 − 2(cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽).</p> Signup and view all the answers

<p>The basic identity applied is cos²θ + sin²θ = 1.</p> Signup and view all the answers

<p>It is treated as −sin(𝛽) due to the properties of the sine function.</p> Signup and view all the answers

<p>sin^(-1)x can refer to the inverse sine function or to csc(x), while arcsin(x) specifically denotes the inverse sine function with restricted domains.</p> Signup and view all the answers

<p>They are not pure inverses as they only apply to outputs within restricted domains, which ensures unique values.</p> Signup and view all the answers

<p>arcsin(sin(7)) = 7 - 2π because 7 is outside the acceptable range of the arcsin function.</p> Signup and view all the answers

<p>sin(arcsin(7)) = DNE (Does Not Exist) because 7 exceeds the range of the sine function (-1 to 1).</p> Signup and view all the answers

<p>The range of arcsin is restricted to $[-π/2, π/2]$.</p> Signup and view all the answers

<p>The equation holds true when b = 7 - 2π, since this value lies within the arcsin range.</p> Signup and view all the answers

<p>An angle is valid for an inverse trigonometric function if it falls within the function's specific range.</p> Signup and view all the answers

<p>csc(x) is the reciprocal of sin(x), defined as 1/sin(x).</p> Signup and view all the answers

<p>It has a removable discontinuity at $x = 0$.</p> Signup and view all the answers

<p>If $f$ is a continuous function on the interval $[a, b]$, then it takes every value between $f(a)$ and $f(b)$.</p> Signup and view all the answers

<p>Because a continuous function cannot be broken, meaning you cannot lift your pencil while drawing it.</p> Signup and view all the answers

<p>At $f(0) = -1$ and $f(\frac{\pi}{2}) = \frac{\pi}{2}$, by the IVT, since $-1 &lt; 0 &lt; \frac{\pi}{2}$, there is a $c$ where $f(c) = 0$.</p> Signup and view all the answers

<p>The IVT ensures that for every positive $eta$ between $0$ and $\zeta$, there is a corresponding $c$ that satisfies $c^n = \beta$.</p> Signup and view all the answers

<p>$f(x)$ is continuous since both $x$ and $\cos(x)$ are continuous functions, enabling the application of the IVT.</p> Signup and view all the answers

<p>The limits as $x$ approaches 0 highlight the removable discontinuity and indicate the limit is 1.</p> Signup and view all the answers

<p>It allows us to show that functions attain certain values over an interval, which is fundamental in analysis.</p> Signup and view all the answers

<p>The general formula is given by $y - y_0 = k(x - x_0)$.</p> Signup and view all the answers

<p>It begins by selecting another point, x, and calculating the slope of the line through (a, f(a)) and (x, f(x)).</p> Signup and view all the answers

<p>The purpose is to obtain the slopes of the tangent lines that converge towards the slope at x = a.</p> Signup and view all the answers

<p>The difference quotient represents the slope of the secant line, calculated as $\frac{f(x) - f(a)}{x - a}$.</p> Signup and view all the answers

<p>It denotes taking the limit of the difference quotient as x approaches the point a.</p> Signup and view all the answers

<p>It ensures that the tangent line can be accurately evaluated at that specific point.</p> Signup and view all the answers

<p>The derivative is expressed as $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$.</p> Signup and view all the answers

<p>It implies that as x approaches a, the calculated slopes converge to the exact slope of the tangent line at that point.</p> Signup and view all the answers

<p>The derivative is 𝑓 ′(𝑥) = lim (𝑐 - 𝑐)/ℎ as ℎ approaches 0, which simplifies to 0.</p> Signup and view all the answers

<p>The derivative is calculated as 𝑓 ′(𝑥) = lim (ℎ)/(ℎ) as ℎ approaches 0, resulting in 1.</p> Signup and view all the answers

<p>𝑓 ′(𝑎) = lim (cos(𝑎 + ℎ) - cos(𝑎))/ℎ as ℎ approaches 0 = -sin(𝑎).</p> Signup and view all the answers

<p>The limit process helps determine the instantaneous rate of change, which the derivative represents, by considering values arbitrarily close to a point.</p> Signup and view all the answers

<p>The negative sign indicates that the slope of the cosine function is decreasing, which aligns with its behavior as it transitions from positive to negative values.</p> Signup and view all the answers