Question

In Exercises 29 and 30, describe and correct the error in finding the inverse of the function. $$ \begin{aligned} f(x) &=\frac{1}{7} x^2, x \geq 0 \\ y &=\frac{1}{7} x^2 \\ x &=\frac{1}{7} y^2 \\ 7 x &=y^2 \\ \pm \sqrt{7 x} &=y \end{aligned} $$

Short answer

The error is in the step where 'x' and 'y' were switched. The correct equation should be \(x = 7y^2\) not \(x = \frac{1}{7} y^2\). The correct inverse of function \(f(x) = \frac{1}{7} x^2 , x \geq 0\) is \(y = \sqrt{\frac{x}{7}} \).

Step-by-step solution

Identify the Error

The exercise involves the steps in finding the inverse of function \(f(x)\). The given steps show that the equation \(x = \frac{1}{7} y^2\) was used in place of \(y = \frac{1}{7} x^2\). The 'x' and 'y' positions were not switched correctly.

Correct the Error

To find the inverse of function \(f(x)\), the 'x' and 'y' positions need to be switched in the equation \(y = \frac{1}{7} x^2 \). This will result in the equation \(x = \frac{1}{7} y^2 \). However, this is not what was done in the exercise. The correct equation should be \(x = 7 y^2 \).

Solve for y

To find the inverse function, the equation needs to be solved for 'y'. Therefore, starting with the corrected equation \(x = 7y^2 \), dividing both sides by 7 results in \( y^2 = \frac{x}{7} \). Taking the square root of both sides gives the solution \( y = \sqrt{\frac{x}{7}} \)

Verify the result

To verify the correctness of the inverse function, it can be compared with initial function. It can be observed that the inverse function \( y = \sqrt{\frac{x}{7}} \) will yield 'x' when squared and multiplied by 7, hence it is the correct inverse function.

Key concepts

Function Composition

Function composition is when you take two functions and apply one after the other. Essentially, you are plugging one function into another. This is often denoted as \((f \circ g)(x) = f(g(x))\), where function \(g(x)\) is input into function \(f(x)\). For students, it's crucial to visualize this as a process of steps, where the output of one function becomes the input for another.

• Start with the inner function: Calculate \(g(x)\).
• Then use the result where \(f(x)\) needs an input.

When dealing with inverse functions, composition helps verify if functions really are inverses of each other. If \(f(x)\) and \(f^{-1}(x)\) are true inverses, composing them should give the identity function: \((f \circ f^{-1})(x) = x\) and \((f^{-1} \circ f)(x) = x\). This checks whether each function undoes the effect of the other.

Quadratic Functions

A quadratic function is a polynomial of degree 2, usually given as \(f(x) = ax^2 + bx + c\). In this task, the quadratic function provided is \(f(x) = \frac{1}{7} x^2\). This form is quite special since it simplifies the handling of quadratic relationships.
Quadratics always form a parabolic shape when graphed, opening upwards if the coefficient \(a\) is positive and downwards if negative. The vertex form of a quadratic can be helpful for finding these min or max values and the "turning point" of the graph. For functions such as \(\frac{1}{7} x^2\), understanding that the coefficient in front of \(x^2\) affects the "width" of the parabola is key.

• Smaller coefficients make the parabola wider.
• Larger coefficients squish it inward.

Remember, quadratic functions with positive \(x^2\) terms must always pass through the origin if they lack constant and linear terms, as in this problem.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or to derive specific solutions, such as when finding inverses. In the given exercise, error correction demonstrated crucial algebraic steps. Here's how:

• Switch variables: The typical first step to finding an inverse mathematically is to swap the roles of \(x\) and \(y\). This essentially asks, "For which \(y\) does this \(x\) occur?"
• Isolate \(y\): After switching, solve systematically to isolate \(y\). From \(x = 7y^2\), divide by 7 and apply the square root. Carefully handle square roots, especially for functions restricted to \(x \geq 0\), ensuring the output also reflects valid results.

The goal is to conduct these manipulations cleanly to find a new expression in terms of \(y\). The final goal is a new function that directly maps \(x\) to \(y\), verifying that both function and its inverse undo each other's results.