Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Common Functions

Calculus

0. Functions

Common Functions

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2:09 minutes

Problem 1.56

Textbook Question

Finding inverses Find the inverse function.

ƒ(x) = 3x - 4

Verified step by step guidance

Step 1: Understand the problem. We need to find the inverse of the function $ f(x) = 3x - 4 $. The inverse function, denoted as $ f^{-1}(x) $, is a function that 'reverses' the effect of $ f(x) $.

Step 2: Replace $ f(x) $ with $ y $. So, we have $ y = 3x - 4 $.

Step 3: Swap $ x $ and $ y $ to find the inverse. This gives us $ x = 3y - 4 $.

Step 4: Solve for $ y $. Add 4 to both sides to get $ x + 4 = 3y $.

Step 5: Divide both sides by 3 to isolate $ y $. This gives $ y = \frac{x + 4}{3} $. Thus, the inverse function is $ f^{-1}(x) = \frac{x + 4}{3} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y as input and returns x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate variables. To find the inverse of a function, you typically start by replacing f(x) with y, then solve for x in terms of y. This process often requires skills such as adding, subtracting, multiplying, and dividing both sides of the equation.

Function Notation

Function notation is a way to denote functions and their outputs clearly. In this context, f(x) represents the output of the function for a given input x. Understanding function notation is crucial for identifying the original function and correctly expressing its inverse, typically denoted as f⁻¹(x).

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Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f (x) = \frac{1}{x}$ , then $f^{- 1} (x) = \frac{1}{x}$

Textbook Question

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).

$f\left(x\right)=\ln\left(3x+1\right)$

Textbook Question

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).

$f\left(x\right)=10^{-2x}$

Textbook Question

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).

$f\left(x\right)=\frac{x}{x-2}$ , for $x\gt{2}$

Textbook Question

Finding inverses Find the inverse function.

ƒ(x) = 3x² + 1, for x ≤ 0

Textbook Question

Inverse of composite functions

a. Let g(x) = 2x + 3 and h(x) = x³. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Textbook Question

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Textbook Question

Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure) <IMAGE>.

a. Find the domain and a formula for each function.

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Finding inverses Find the inverse function.ƒ(x) = 3x - 4

Key Concepts

Inverse Functions

Algebraic Manipulation

Function Notation

Watch next

Finding inverses Find the inverse function.

ƒ(x) = 3x - 4