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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 1.1.1
Capitolo 1, Problema 1.1.1

Functions


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


f(x) = 1 + x²

Guida verificata passo dopo passo
1
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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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.
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Finding the Domain and Range of a Graph

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, ∞).
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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).
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Introduction to Polynomial Functions
Pratica correlata
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Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = x³/8

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In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x² + 1

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Functions and Graphs


Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.

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In Exercises 39–42, express the given quantity in terms of sin x and cos x.


sin (2π − x)

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General Sine Curves


For


f(x) = A sin ((2π/B)(x – C) +D


identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.


y = L/2π sin (2πt/L), L > 0

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In Exercises 9–16, determine whether the function is even, odd, or neither.


𝔂 = x⁴ + 1

x³ - 2x

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