Math Problem Statement

Знайдіть область значень функції y = x + 3, заданої на проміжку: a) [-3; 3]; b) [1; 7]; в) [0; +∞).

Solution

The problem in the image asks to find the range ("область значень") of the function y=x+3y = x + 3, given different domains of xx.

Let's go through each part:

Function:

y=x+3y = x + 3

Part (a): Domain [3;3][-3; 3]

If xx is between 3-3 and 33, we substitute these values into the function:

  • When x=3x = -3, y=3+3=0y = -3 + 3 = 0
  • When x=3x = 3, y=3+3=6y = 3 + 3 = 6

Thus, the range is [0;6][0; 6].

Part (b): Domain [1;7][1; 7]

For xx between 11 and 77, we calculate:

  • When x=1x = 1, y=1+3=4y = 1 + 3 = 4
  • When x=7x = 7, y=7+3=10y = 7 + 3 = 10

So, the range is [4;10][4; 10].

Part (c): Domain [0;+)[0; +\infty)

For x0x \geq 0, the function becomes:

  • When x=0x = 0, y=0+3=3y = 0 + 3 = 3
  • As xx increases, yy increases without bound.

Thus, the range is [3;+)[3; +\infty).


Would you like more details on any part of the solution?

Here are 5 related questions to consider:

  1. What would the range be if the domain was (;0](-\infty; 0]?
  2. How does shifting the function y=x+3y = x + 3 affect the range compared to y=xy = x?
  3. What happens to the range if we change the function to y=x3y = x - 3?
  4. Can the range of this linear function ever be restricted to a finite interval?
  5. How does the slope of the function y=x+3y = x + 3 influence the range?

Tip: For linear functions like y=mx+by = mx + b, the slope mm and the intercept bb define how the function behaves over any domain, making range calculations straightforward.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Domain and Range

Formulas

y = x + 3

Theorems

Basic properties of linear functions

Suitable Grade Level

Grades 8-10