Question

\(g: R \rightarrow R, g(x)=3+3 \sqrt{x}\) and \(f(g(x))=2-3 \sqrt{x}+x\) then \(\mathrm{f}(\mathrm{x})=\ldots\) (a) \(x^{3}-x^{2}+x-5\) (b) \(x^{3}-9 x^{2}+26 x-22\) (c) \(x^{3}+9 x^{2}-26 x+5\) (d) \(x^{3}+x^{2}-x+5\)

Short answer

\(f(x) = -x + 5 + \frac{(x-3)^2}{9}\)

Step-by-step solution

Find the inverse of g(x)

To find the inverse of a function, we need to switch the roles of x and g(x), and then solve for the new input \(x\). So for the function \(g(x) = 3 + 3\sqrt{x}\), we will replace \(g(x)\) with \(y\). \(y = 3 + 3\sqrt{x}\) Now, switch the roles of \(x\) and \(y\): \(x = 3 + 3\sqrt{y}\) Now, let's solve for \(y\).

Solve for y

To solve for \(y\), we need to isolate it on one side of the equation. Start by subtracting 3: \(x - 3 = 3\sqrt{y}\) Next, divide by 3: \(\frac{x-3}{3} = \sqrt{y}\) Now, square both sides of the equation: \((\frac{x-3}{3})^2 = y\) This simplifies to: \(y = \frac{(x-3)^2}{9}\) Now that we have the inverse function, we can denote it as: \(g^{-1}(x) = \frac{(x-3)^2}{9}\)

Substitute the inverse function into f(g(x))

Now we'll replace \(g(x)\) with its inverse function, \(g^{-1}(x)\), in the expression for \(f(g(x))\): \(f(g(x)) = 2 - 3\sqrt{x} + x \Rightarrow f(g^{-1}(x)) = 2 - 3\sqrt{\frac{(x-3)^2}{9}} + \left(\frac{(x-3)^2}{9}\right)\) Since we want to find \(f(x)\), we'll just replace \(g^{-1}(x)\) with \(x\), and we'll have: \(f(x) = 2 - 3\sqrt{\frac{(x-3)^2}{9}} + \frac{(x-3)^2}{9}\)

Simplify the expression for f(x)

Let's simplify the expression for \(f(x)\): \(f(x) = 2 - 3\left(\frac{x-3}{3}\right) + \frac{(x-3)^2}{9}\) \(f(x) = 2 - (x-3) + \frac{(x-3)^2}{9}\) \(f(x) = -x + 5 + \frac{(x-3)^2}{9}\) Now let's compare this to the given options (a), (b), (c), and (d).

Comparison

Comparing \(f(x) = -x + 5 + \frac{(x-3)^2}{9}\) with the given options, we can see that none of the options match the obtained expression for \(f(x)\). This means that there might be an issue with the given options or the question might have some errors. In this case, the given options do not provide a correct answer to the problem.

Key concepts

Function Composition

Function composition involves creating a new function by plugging one function into another. In this exercise, we are dealing with two functions: \( g(x) = 3 + 3\sqrt{x} \) and \( f(g(x)) = 2 - 3\sqrt{x} + x \). Function composition can be thought of as a process where the output of the first function becomes the input for the second function.
When composing these functions, we often look for the expression of the form \( f(g(x)) \). This means that when you take a value for \( x \), you first apply \( g \), and the resulting output becomes the input for \( f \).
The task here was to replace \( g(x) \) with its inverse in the expression of \( f \), highlighting the intricate relationship between functions required to solve for \( f(x) \). Understanding this relationship helps enhance analytical skills in dealing with composite functions.

Simplifying Expressions

Simplifying expressions is crucial for making complex mathematical problems more manageable. The essence of simplifying lies in expressing functions in their simplest form, which often involves combining like terms or performing arithmetic operations to condense expressions.
In the problem, we were required to simplify the expression \( f(x) = 2 - 3\sqrt{\frac{(x-3)^2}{9}} + \frac{(x-3)^2}{9} \). This starts by simplifying the square root term \( 3\sqrt{\frac{(x-3)^2}{9}} \), which reduces to \( x-3 \) after canceling out the square and the division.

\( f(x) = 2 - (x-3) + \frac{(x-3)^2}{9} \)

• First, distribute the negative sign across the \( (x-3) \).
• The expression becomes \( f(x) = -x + 5 + \frac{(x-3)^2}{9} \).

By simplifying the function step by step, we extract a concise and clear expression, enabling easier comparison and evaluation.

Comparing Polynomial Functions

Polynomial functions, characterized by their series of powers of \( x \), often need to be compared to identify similarities and differences in their coefficients or structures. In algebra and calculus, comparisons reveal insights about the behavior of functions, such as intersections, roots, and relative sizes of values.
After simplifying \( f(x) \) in the exercise, it was necessary to compare it against the provided polynomial options to find a match. We observe:

• Look at the structure of each polynomial function option provided: each is a third-degree polynomial.
• Examine the terms of \( f(x) \) individually and compare them with each option.

The process of elimination in comparing against options is crucial for identifying any mismatch or errors in either the provided options or the original problem. The lack of a matching option in this task suggests a possible misalignment, an error in the question, or an oversight in the expression of \( f(x) \) during simplification.