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

Introduction to Functions

Calculus

0. Functions

Introduction to Functions

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2:26 minutes

Problem 1.1.1

Textbook Question

Functions

In Exercises 1–6, find the domain and range of each function.

f(x) = 1 + x²

Verified step by step guidance

Step 1: Understand the function f(x) = 1 + x². This is a quadratic function, which is a type of polynomial function.

Step 2: Determine the domain of the function. Since f(x) = 1 + x² is a polynomial function, it is defined for all real numbers. Therefore, the domain is all real numbers, which can be expressed as (-∞, ∞).

Step 3: Analyze the range of the function. The expression x² is always non-negative, meaning it is greater than or equal to 0 for all real x.

Step 4: Since x² is non-negative, the smallest value of f(x) = 1 + x² occurs when x² = 0, which gives f(x) = 1. Therefore, the range starts at 1.

Step 5: As x increases or decreases without bound, x² becomes very large, and consequently, f(x) = 1 + x² also becomes very large. Thus, the range of the function is [1, ∞).

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Key Concepts

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

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function f(x) = 1 + x², the domain includes all real numbers since there are no restrictions on x that would make the function undefined.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. In the case of f(x) = 1 + x², the output is always greater than or equal to 1, as x² is non-negative. Thus, the range is [1, ∞).

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. The function f(x) = 1 + x² is a specific type of quadratic function where a = 1, b = 0, and c = 1, which opens upwards and has a vertex at the point (0, 1).