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Chapter 0: Problem 14

Find a formula for \(f^{-1}(x)\). \(f(x)=5 /\left(x^{2}+1\right), x \geq 0\)

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = \sqrt{\frac{5 - x}{x}}\).

Step by step solution

01

Define the Inverse Function

To find the inverse, we need to switch the roles of \(x\) and \(y\). Start by expressing \(f(x)\) in terms of \(y\), where \(f(x) = y\). So, write: \(y = \frac{5}{x^2 + 1}\).
02

Solve for x in terms of y

To find \(x\) in terms of \(y\), we need to solve the equation \(y = \frac{5}{x^2 + 1}\) for \(x\). Start by multiplying both sides by \(x^2 + 1\): \(y (x^2 + 1) = 5\).
03

Separate x terms

Next, expand the left side: \(yx^2 + y = 5\). Move the \(y\) to the right side to isolate the term with \(x^2\): \(yx^2 = 5 - y\).
04

Isolate x虏

Divide both sides by \(y\) to solve for \(x^2\): \(x^2 = \frac{5 - y}{y}\).
05

Solve for x

Since \(x \geq 0\), take the positive square root to solve for \(x\): \(x = \sqrt{\frac{5 - y}{y}}\).
06

Express f^{-1}(x)

Finally, express the inverse function by switching \(x\) back for \(y\). Therefore, \(f^{-1}(x) = \sqrt{\frac{5 - x}{x}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is often the first step in finding an inverse function. It involves rearranging and simplifying expressions to isolate a variable on one side of the equation. When we find an inverse function, we essentially reverse the operations of the original function to get back to our input value. Let鈥檚 dive into why this is important and how it works:
  • Switching Variables: Start with switching the dependent (\(y\)) and independent (\(x\)) variables in the function. This effectively sets up the equation that the inverse function must satisfy.
  • Rearranging: Manipulate the equation to solve for the original independent variable in terms of the new independent variable. This typically involves algebraic manipulations, such as adding, subtracting, multiplying, and dividing both sides of the equation by the same numbers or expressions.
  • Isolating Variables: Carefully isolate the new dependent variable to express the function explicitly. This requires logical steps to keep the equation accurate.
These steps ensure that we achieve a precise and correct formula for the inverse function. It's like retracing a path 鈥� each operation is reversed to find where it begins.
Square Roots
Square roots play a crucial role in functions, especially when solving equations involving quadratic terms. When you encounter terms like \(x^2\), solving for \(x\) generally involves taking the square root of both sides of an equation.
  • Positive and Negative Roots: A quadratic equation like \(x^2 = a\) has two solutions: \(x = \sqrt{a}\) and \(x = -\sqrt{a}\). When dealing with inverse functions, especially those that have domain restrictions, we choose the relevant root that satisfies the criteria given in the problem.
  • Domain Restrictions: In our initial function, \(x \geq 0\), suggests that we only consider the positive square root in our solution for the inverse function. This restriction is essential for maintaining the function's proper domain in real-world contexts.
Understanding the role of square roots helps in addressing and solving parts of the equation that involve quadratic terms, ensuring accuracy in inverse function derivation.
Function Transformation
Function transformation is the process of applying changes like translations, stretches, or reflections to the graph of a function. It's essential to understand how these transformations affect both the original and inverse functions you encounter.
  • Applying Transformations: Original functions might be transformed through operations such as adding constants (vertical shifts), multiplying by a factor (vertical or horizontal stretches), or taking reciprocals (flips). These affect how the inverse function should be derived.
  • Reflecting Over the Line \(y = x\): Graphically, the inverse of a function is a reflection over the line \(y = x\). This visual representation is a handy way to understand and verify if we have correctly identified the inverse function. If you plot both the function and its inverse, they should be mirror images in this line.
  • Understanding Impact: Recognizing how changes to an original function affect its inverse is key for controlling and predicting the behavior of both functions. For example, if a function is shifted upwards, this will influence how its inverse is expressed.
Grasping these transformations enhances comprehension of inverse functions and prepares students to handle complex function manipulations effortlessly.

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Most popular questions from this chapter

Express \(f\) as a composition of two functions; that is, find \(g\) and \(h\) such that \(f=g \circ h .\) [Note: Each exercise has more than one solution.] (a) \(f(x)=\frac{1}{1-x^{2}}\) (b) \(f(x)=|5+2 x|\)

The number of hours of daylight on a given day at a given point on the Earth's surface depends on the latitude \(\lambda\) of the point, the angle \(\gamma\) through which the Earth has moved in its orbital plane during the time period from the vernal equinox (March 21), and the angle of inclination \(\phi\) of the Earth's axis of rotation measured from ecliptic north \(\left(\phi \approx 23.45^{\circ}\right)\). The number of hours of daylight \(h\) can be approximated by the formula $$ h=\left\\{\begin{array}{ll} 24, & D \geq 1 \\ 12+\frac{2}{15} \sin ^{-1} D, & |D|<1 \\ 0, & D \leq-1 \end{array}\right. $$ where $$ D=\frac{\sin \phi \sin \gamma \tan \lambda}{\sqrt{1-\sin ^{2} \phi \sin ^{2} \gamma}} $$ and \(\sin ^{-1} D\) is in degree measure. Given that Fairbanks, Alaska, is located at a latitude of \(\lambda=65^{\circ} \mathrm{N}\) and also that \(\gamma=90^{\circ}\) on June 20 and \(\gamma=270^{\circ}\) on December 20, approximate (a) the maximum number of daylight hours at Fairbanks to one decimal place (b) the minimum number of daylight hours at Fairbanks to one decimal place.

True-False Determine whether the statement is true or false. Explain your answer. The graph of an even function is symmetric about the \(y\) -axis.

(i) Use a graphing utility to graph the equation in the first quadrant. [Note: To do this you will have to solve the equation for \(y\) in terms of \(x .]\) (ii) Use symmetry to make a hand-drawn sketch of the entire graph. (iii) Confirm your work by generating the graph of the equation in the remaining three quadrants. $$ 9 x^{2}+4 y^{2}=36 $$

True-False Determine whether the statement is true or false. Explain your answer. Curves in the family \(y=-5 \sin (A \pi x)\) have amplitude 5 and period \(2 /|A|\).
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