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Chapter 0: Problem 39

Find the inverse of each function. $$ f(x)=-\frac{1}{x} $$

Short Answer

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

Step by step solution

01

Replace f(x) with y

Start by replacing the function notation \( f(x) \) with \( y \). This gives us the equation \( y = -\frac{1}{x} \).
02

Swap x and y

To find the inverse, swap the positions of \( x \) and \( y \). The equation becomes \( x = -\frac{1}{y} \).
03

Solve for y

Solve the equation for \( y \). Multiply both sides by \( y \) to get: \( xy = -1 \). Then, solve for \( y \) by dividing both sides by \( x \): \( y = -\frac{1}{x} \).
04

Replace y with f^{-1}(x)

Replace \( y \) with \( f^{-1}(x) \) to denote the inverse function. The inverse function is: \( f^{-1}(x) = -\frac{1}{x} \).

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Key Concepts

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

Trigonometry Functions
Trigonometry functions, like sine and cosine, are vital in math. These functions describe the relationship between angles and sides in triangles. Inverse trigonometric functions, like arcsine and arccosine, reverse this relationship. For instance, if you know the sine of an angle, you can find the angle using arcsine. This concept is similar to finding the inverse in algebraic functions.
Understanding trigonometric functions helps in many areas of math and science. It is especially useful when dealing with waves and oscillations, which are common in physics and engineering.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations and expressions to solve for unknowns. In our exercise, we used basic algebra to find the inverse function. Here鈥檚 a breakdown:
  • We started by rewriting the function: Replace \(f(x)\) with \(y\).
  • Then, we swapped \(x\) and \(y\): \(x = -\frac{1}{y}\)
  • Next, we solved for \(y\) by multiplying both sides by \(y\): \(xy = -1\)
  • Finally, we divided by \(x\) to isolate \(y\): \(y = -\frac{1}{x}\)

These steps are common in solving many types of problems in algebra. Practice makes perfect!
Function Notation
Function notation, like \(f(x)\), is a way to represent functions. It shows how input values (x) map to outputs (f(x)). For example, \(f(x) = -\frac{1}{x}\). This means that for any given \(x\), the function will output the negative reciprocal.
When finding inverses, function notation changes slightly. If \(f(x)\) maps \(x\) to \(y\), the inverse function, \(f^{-1}(x)\), maps \(y\) back to \(x\). In our exercise, we found: \(f(x) = -\frac{1}{x}\) and \(f^{-1}(x) = -\frac{1}{x}\). This shows that both the function and its inverse are the same!
Solving Equations
Solving equations is a fundamental skill in math. It involves finding the value of unknowns that satisfy the equation. In our exercise, the key steps were:
  • Writing \(y = -\frac{1}{x}\)
  • Swapping \(x\) and \(y\): \(x = -\frac{1}{y}\)
  • Solving for \(y\) by multiplying both sides by \(y\): \(xy = -1\)
  • Isolating \(y\) by dividing by \(x\): \(y = -\frac{1}{x}\)

These steps show how algebra and inverse functions work together. Practice with different equations to become more comfortable with these techniques.

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Most popular questions from this chapter

Solve each problem. Making Radios The number of radios shipped by ABC Electronics per week to its factory stores is given by \(S(x)=0.90 x,\) where \(x\) is the number manufactured. The number of radios purchased per week is given by \(P(x)=0.80 x\), where \(x\) is the number shipped. Write the number purchased per week as a function of the number manufactured.

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ g(-x) $$

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ f(-4) $$

Working in small groups, plot the points \((-1,3)\) and \((4,1)\) on a sheet of graph paper. Assuming that these two points are adjacent vertices of a square, find the other two vertices. Now select your own pair of adjacent vertices and "complete the square." Next, start with the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) as adjacent vertices of a square and write expressions for the coordinates of the other two vertices.

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ t, \text { if } f(t)=10 $$
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