find the domain and range of the following as an inequality and interval notation.
y=sqrtx-3
2: y=3x+5
3: y= 2/x-7
4; y=2+sqrt x
domain = set of all possible values of x
range = set of all possible values of y
therefore,
(1) y = sqrt(x-3)
note that the term inside the sqrt can be zero but cannot be negative, for it will become imaginary. thus,
Domain: x must be greater than or equal to 3.
also, y can be zero (if x=3) but can never be negative since sqrt of any positive real number is always positive, thus,
Range: y must be greater than or equal to 0.
(2) y = 3x + 5
this is an equation of a line, and note that there is neither restriction in both x and y, thus,
Domain: x can be all real numbers
Range: y can be all real numbers
(3) y = 2/(x-7)
note that denominators cannot be 0, thus,
Domain: x can be all real numbers except 7.
for the range, observe that y cannot be equal to zero, because, if so,
0 = 2/(x-7) , then cross-multiplying,
0 = 2 , which is wrong, thus
Range: y is all real numbers except 0
(4) y = 2 + sqrt(x)
note that this is almost similar to #1, except that there is +2. therefore,
Domain: x must be greater than or equal to zero.
for the range, the minimum possible value of x is zero, but because there is +2,
Range: y must be greater than or equal to 2.
hope this helps~ :D
To find the domain and range of the given functions, let's analyze each function separately:
1. y = √(x - 3)
Domain (inequality): Since the square root function is defined for non-negative real numbers or positive infinity, we need to find the values of x that make (x - 3) non-negative:
x - 3 ≥ 0
x ≥ 3
Therefore, the domain of the function is x ≥ 3.
Domain (interval notation): In interval notation, we write the domain as [3, ∞).
Range: Since the square root function always yields non-negative outputs, the range of the function is y ≥ 0.
Range (interval notation): In interval notation, we write the range as [0, ∞).
2. y = 3x + 5
There are no restrictions on x for this linear function, so the domain is all real numbers.
Domain (interval notation): In interval notation, we write the domain as (-∞, ∞).
Range: Since the slope of the line is positive (3), the range of the function is (-∞, ∞).
Range (interval notation): In interval notation, we write the range as (-∞, ∞).
3. y = 2/(x - 7)
Domain (inequality): To find the values of x that make the denominator non-zero, we solve for (x - 7) ≠ 0:
x - 7 ≠ 0
x ≠ 7
Therefore, the domain of the function is x ≠ 7.
Domain (interval notation): In interval notation, we write the domain as (-∞, 7) U (7, ∞).
Range: Since the function is a rational function, the range excludes the values of y where the denominator approaches zero. As x approaches positive infinity or negative infinity, the function approaches zero. Hence, the range is all real numbers except y = 0.
Range (interval notation): In interval notation, we write the range as (-∞, 0) U (0, ∞).
4. y = 2 + √x
Domain (inequality): Since the square root function is defined for non-negative real numbers or positive infinity, there are no restrictions on x. Hence, the domain is all real numbers.
Domain (interval notation): In interval notation, we write the domain as (-∞, ∞).
Range: The square root function always yields non-negative outputs, so the range of the function is y ≥ 2.
Range (interval notation): In interval notation, we write the range as [2, ∞).
To find the domain and range of the given functions, we need to analyze the behavior of each function and identify any restrictions.
1. Function: y = √(x - 3)
Domain:
The square root function (√) is defined for non-negative values. Therefore, x - 3 must be greater than or equal to zero.
x - 3 ≥ 0
x ≥ 3
Range:
The range of the square root function (√) is all real numbers greater than or equal to zero.
Range: [0, ∞)
2. Function: y = 3x + 5
Domain:
This is a linear function, which means there are no restrictions on the x-values.
Domain: (-∞, ∞)
Range:
A linear function has a constant slope, meaning it covers all real numbers.
Range: (-∞, ∞)
3. Function: y = 2/(x - 7)
Domain:
In this case, the denominator (x - 7) cannot equal zero because division by zero is undefined.
x - 7 ≠ 0
x ≠ 7
Range:
The rational function 2/(x - 7) takes all possible real values except when x equals 7 (due to the denominator being zero).
Range: (-∞, 7) U (7, ∞)
4. Function: y = 2 + √x
Domain:
Similar to the first function, the square root function (√x) requires non-negative values.
x ≥ 0
Range:
The range of the square root function (√) is all real numbers greater than or equal to zero.
Range: [2, ∞)