Question

\(f(x)=\sin ^{-1}\left\\{4-(x-7)^{3}\right\\}^{(1 / 5)}\) then \(f^{-1}(x)=\ldots .\) (a) \(\left(7-\sin ^{5} x\right)^{(1 / 3)}\) (b) \(7+\left(4-\sin ^{5} x\right)^{(1 / 3)}\) (c) \(7+\left(4+\sin ^{5} \mathrm{x}\right)^{(1 / 3)}\) (d) \(\left(7+\sin ^{5} x\right)^{(1 / 3)}\)

Short answer

The short version of the answer is: \(f^{-1}(x) = 7 + \left\\{4-\sin^5(x)\right\\}^{(1/3)}\)

Step-by-step solution

Analyze the given function

First, let's examine the given function: \(f(x)=\sin^{-1}\left\\{4-(x-7)^{3}\right\\}^{(1 / 5)}\). The objective is to find its inverse function.

Define inverse function

Let's begin by defining the inverse function as \(f^{-1}(x) = y\), and let the given function be equal to x.

Substitute inverse function

Now, we need to rewrite the original equation of f, replacing it with its inverse function. We should replace \(f^{-1}(x)\) with \(y\) and replace \(f(x)\) with \(x\): \(x = \sin^{-1}\left\\{4-(y-7)^{3}\right\\}^{(1 / 5)}\)

Applying inverse trigonometric function

To find the inverse of the given function, we need to reverse the equation and solve for y. Since we have the sin inverse terms, we need to take the sine of both sides: \(\sin(x) = \left\\{4-(y-7)^3\right\\}^{(1/5)}\)

Raise both sides to the power of 5

Now, raise both sides to the power of 5 to isolate the cubic equation: \((\sin(x))^5 = 4-(y-7)^3\)

Solve the cubic equation

Now, rearrange the equation to solve for the cubic part: \((y-7)^3 = 4-\sin^5(x)\) Then, take the cubic root of both sides: \(y-7 = \left\\{4-\sin^5(x)\right\\}^{(1/3)}\) Finally, solve for y: \(y = 7 + \left\\{4-\sin^5(x)\right\\}^{(1/3)}\) Looking at our options, we can see that this matches option (b): \(f^{-1}(x) = 7 + \left\\{4-\sin^5(x)\right\\}^{(1/3)}\)

Key concepts

Function Inverse

When you hear the term "function inverse," think of reversing actions. If you have a function, finding its inverse essentially means figuring out how to revert its effect. For example, if a function takes an input and adds 5, its inverse will subtract 5. For mathematical functions, we often denote the inverse function with a superscript (-1), such as \( f^{-1}(x) \). To find the inverse, follow these general steps:

• Replace the function notation with "y." This step is like saying \( y = f(x) \).
• Swap "x" and "y." Your equation will now look like \( x = f(y) \).
• Solve this new equation for "y." The result is your inverse function.

These steps are exactly what we did in the original exercise. This method allows you to "undo" operations in the given function to return to the original input, thereby defining the inverse function.

Cubic Equations

Cubic equations are equations where the variable has an exponent of 3, making them polynomials of degree three. A common form looks like \( ax^3 + bx^2 + cx + d = 0 \). Solving cubic equations requires different techniques than simpler polynomials (like linear or quadratic). Here are a few steps that might be used.

• **Factoring:** Try to factor the polynomial. This is not always possible, but when it is, it's often the simplest method to solve the equation.
• **Cubic Formula:** Similar to the quadratic formula but more complex, it can solve any cubic equations, though it's often not used because of the complexity involved.

In the exercise, our cubic equation took on the form \((y - 7)^3\). The next step was solving for \(y\), which required us to take the cube root. Understanding how to handle cube roots helps in isolating variables and solving these types of equations efficiently.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable. These identities are tools that simplify the expressions and solve trigonometric equations. Some basic examples include:

• **Pythagorean identities:** Relationships like \( \sin^2(x) + \cos^2(x) = 1 \).
• **Angle sum identities:** Which help expand expressions like \( \sin(x + y) \) or \( \cos(x + y) \).

In the given problem, we used the identity \( \sin^5(x) \) to manipulate and solve for the inverse of the function. Understanding how to use these identities properly can simplify complex trigonometric problems. Knowing these identities inside out can often turn convoluted tasks into straightforward problems, improving problem-solving speed and accuracy.