Inverse Functions and Angle Formulas

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Questions and Answers

What are the two general steps to find the inverse function of a given function?

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

Why does the function 𝑦 = 𝑥^2 not have an inverse?

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

What is the 'Horizontal Line Test' and why is it significant?

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

Under what conditions does a function have an inverse?

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

If given y = f(x), how can you determine if f has an inverse?

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

What is the relationship between the uniqueness of outputs and the existence of an inverse function?

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

How can you find the inverse of the function y = x^3 - 2?

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

What happens if two different x-values produce the same output in a function?

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

What is the final expression for cos(𝛼 − 𝛽) derived from the equation presented?

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

How does the expansion of [cos(𝛼 − 𝛽) − 1]² relate to the identity of cosine?

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

What formula is derived for the sum of angles from the difference of angles formula?

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

What leads to the simplification of the right-hand side during the expansion?

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

Identify the role of the negative sign in the angle sum formula proof.

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

What do the terms 2 − 2 cos(𝛼 − 𝛽) equate to when manipulated?

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

What basic identity is applied during the proof of the sum of angles formula?

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

How is the term sin(−𝛽) treated in the context of the angle addition formula?

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

Explain the difference between sin^(-1)x and arcsin(x).

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. Signup and view all the answers

Why are the functions arcsin, arccos, arctan, and arccot not pure inverses of their respective sine, cosine, tangent, and cotangent functions?

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

Compute arcsin(sin(7)). What is the reasoning behind your answer?

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

What happens when you attempt to compute sin(arcsin(7))?

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

Define the range of values of the arcsin function.

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

When does the equation sin(b) = sin(7) hold true for b within the range of arcsin?

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

Illustrate how to determine if an angle is valid for an inverse trigonometric function.

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

For the function csc(x), what is its relationship with sin(x)?

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

What type of discontinuity does the function $f(x) = \frac{\tan(x)}{\sin(x)}$ exhibit at $x = 0$?

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

State the Intermediate Value Theorem in simple terms.

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

Why must you 'hit every point in between' when applying the Intermediate Value Theorem?

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

Show that the function $f(x) = x - \cos(x)$ has a zero in the interval $[0, \frac{\pi}{2}]$. How are the values at the endpoints significant?

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

Explain how the Intermediate Value Theorem proves the existence of a unique $c > 0$ such that $c^n = \zeta$ for $\zeta > 0$.

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

Discuss the significance of continuity in relation to the function $f(x) = x - \cos(x)$.

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

What conclusion can be drawn from the behavior of the function $f(x) = \frac{\tan(x)}{\sin(x)}$ near $x = 0$?

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

Why is the Intermediate Value Theorem considered 'incredibly useful'?

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

What is the general formula for a tangent line at point (a, f(a))?

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

How does the process of finding the slope of a tangent line begin?

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

In the context of derivatives, what is the purpose of moving x closer to a?

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

What does the difference quotient represent in the process of finding a derivative?

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

What is meant by the notation $\lim_{x \to a}$ in the context of derivatives?

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

What is the significance of the function f(x) being defined at point a?

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

How do we mathematically express the derivative of f(x) at a using limits?

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

What does it mean for slopes of tangent lines to get 'closer and closer' to a specific value?

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

Using the definition of the derivative, show how you can calculate the derivative of 𝑓(𝑥) = 𝑐 where 𝑐 is a constant.

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

Explain why the derivative of 𝑓(𝑥) = 𝑥 is equal to 1 for all x.

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

Derive 𝑓 ′(𝑎) for the function 𝑓(𝑎) = cos(𝑎) using the definition of the derivative.

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

Provide a brief explanation of why the limit process is essential in finding the derivative of functions.

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

What is the significance of the negative sign in the derivative of the cosine function?

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. Signup and view all the answers

Flashcards

Inverse function

A function that reverses the effect of another function. If f(x) = y, then the inverse function, denoted by f⁻¹(y) = x.

Finding the inverse function (Step 1)

Express the input variable (x) in terms of the output variable (y).

Finding the inverse function (Step 2)

Swap the input and output variables (x and y).

Horizontal Line Test

A test to determine if a function has an inverse. A function has an inverse if and only if no horizontal line intersects its graph more than once.