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Chapter 4: Problem 2

Write an equation for the inverse function. $$ g(x)=2 x-7 $$

Short Answer

Expert verified
\[ g^{-1}(x) = \frac{x + 7}{2} \]

Step by step solution

01

Replace g(x) with y

To find the inverse of the function, first replace the function notation with y. Given: \[ y = 2x - 7 \]
02

Swap x and y

Interchange the roles of x and y to reflect finding the inverse. This gives: \[ x = 2y - 7 \]
03

Solve for y

Rearrange the equation to solve for y. \[ x + 7 = 2y \] Then divide by 2: \[ y = \frac{x + 7}{2} \]
04

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

Finally, rewrite the equation to show the inverse function. This results in: \[ g^{-1}(x) = \frac{x + 7}{2} \]

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Key Concepts

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

Algebra
Algebra is like a toolset with various techniques to solve mathematical problems. When finding an inverse function, we use algebraic methods to rearrange equations. This often involves moving terms from one side of an equation to the other and isolating specific variables.
For example, consider rearranging the equation:
We start with: \[ y = 2x - 7 \]
Then to find the inverse, we need to move terms around using algebraic rules. By switching x and y, adding 7 to both sides, and then dividing by 2, we can solve for y. These steps showcase algebra in action.
Function Notation
Function notation helps us label and understand functions clearly. When we write something like \( g(x) = 2x - 7 \), we are defining a function named 'g' that changes 鈥榵鈥� into 鈥�2x - 7鈥�.
This simple notation is very powerful. It lets us see how inputs are transformed. When finding an inverse function, the notation changes so that \( g^{-1}(x) \) indicates the inverse function. This tells us how to reverse the original function鈥檚 process. Recognizing and using this notation is key in understanding and communicating about functions and their inverses.
Solving Equations
Solving equations is about finding unknown values. When we solve for y in the inverse function exercise, we do exactly this. Starting from \[ x = 2y - 7 \] we need y by itself.
We add 7 to both sides and get \[ x + 7 = 2y \].
Then, we divide both sides by 2, isolating y: \[ y = \frac{x + 7}{2} \]. This process of isolating the variable shows basic solving techniques.
These techniques also show up in many areas of math and science, making them very valuable.
Inverse Operations
Inverse operations reverse the effects of other operations. For example, if you add 7, you can subtract 7 to get back to the original number. Similarly, multiplying by 2 can be reversed by dividing by 2.
When finding an inverse function, these operations are key. We start with a function, swap x and y, and then use inverse operations to isolate y. This process gives us the inverse function \( g^{-1}(x) \). Recognizing which operations to use and mastering them is vital in math.

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{8}(3 y-5)+10=12\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log \left(p^{2}+6 p\right)=\log 7\)

Show that \(\left(\frac{e^{x}+e^{-x}}{2}\right)^{2}-\left(\frac{e^{x}-e^{-x}}{2}\right)^{2}=1\).

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{7}(12-t)=\log _{7}(t+6)\)

Show that \(\ln \left(\frac{c+\sqrt{c^{2}-x^{2}}}{c-\sqrt{c^{2}-x^{2}}}\right)=2 \ln \left(c+\sqrt{c^{2}-x^{2}}\right)-2 \ln x\)
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