Domain & Range
What numbers are in the domain for:
(-1, 9), (2, 7), (4, 6), (6, 8)
{-1, 2, 4, 6}
What is the GCF of 12 and 18?
The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common factor is 6.
Simplify (2a³b²)⁴.
Apply the power rule to each factor within the parentheses. (2a³b²)⁴ = 2⁴ * (a³)⁴ * (b²)⁴ = 16a¹²b⁸
What is the product of (x + 3) and 5?
5(x + 3) = 5x + 15
Rewrite the equation y = 2x + 1 in standard form.
Subtract 2x from both sides to get -2x + y = 1. (Note: Some instructors might prefer to multiply by -1 to make A positive: 2x - y = -1)
How can you use a graph to determine the domain and range of a function?
The function f(x) = x² represents a parabola that opens upwards. Since x² is always non-negative (0 or positive), the range of this function is all non-negative real numbers, or [0, ∞) in interval notation.
Find the GCF of 24, 36, and 48.
Listing all factors can be time-consuming for larger numbers. A more efficient method is to find the prime factorization of each number:
Simplify (x⁻²)³.
A negative exponent means to take the reciprocal. (a³/b²)⁻¹ = b²/a³
Multiply (x + 2)(x + 1)
x² + x + 2x + 2 = x² + 3x + 2
The equation of a line is 3x + 6y = 12. Write an equivalent equation in standard form with the smallest possible integer coefficients.
Divide the entire equation by 3 to get x + 2y = 4
Recall the definition of a function.
A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range.
The first set is called the domain of the function.
The second set is called the range of the function.
Find the GCF of 15 and 25.
15 = 3 x 5; 25 = 5 x 5. The only common prime factor is 5, so the GCF is 5.
Simplify (4x⁻²y³/2x³y⁻¹)².
First simplify inside the parentheses: (4x⁻²y³/2x³y⁻¹) = 2x⁻⁵y⁴. Then square the result: (2x⁻⁵y⁴)² = 4x⁻¹⁰y⁸ = 4y⁸/x¹⁰
Expand (2x - 3)(x + 4)
2x² + 8x - 3x - 12 = 2x² + 5x - 12
A line passes through (2, 1) and has a slope of -1/2. Write the equation of the line in standard form.
First use point-slope form: y - 1 = -1/2(x - 2). Then convert to standard form: x + 2y = 4
The daily cost of hiring a plumber, y to work x hours on a repair project can be modeled using a linear function.The plumber charges $45 per hour. The plumber works a maximum of 8 hours per day.
For one day of work, what is the range of the function for this situation?
80<y<440
Find the GCF of 72 and 90 using the Euclidean Algorithm.
The Euclidean Algorithm is a more efficient method for larger numbers.
Simplify [(x⁴y⁻²)⁻¹(x⁻¹y²)³] / (x³y⁻¹)⁻².
This involves multiple steps of applying the various exponent rules. First, simplify each term in the numerator individually, then simplify the denominator. Then, divide the numerator by the denominator, remembering to subtract exponents of the same base. The final simplified expression will be x⁻⁷y¹⁰
Simplify (3x + 2y)(x - 4y)
3x² - 12xy + 2xy - 8y² = 3x² - 10xy - 8y²
Two lines are parallel. One has the equation 2x - 5y = 10. What can you say about the equation of the other line in standard form?
Parallel lines have the same slope. The slope of 2x - 5y = 10 is 2/5. The other line's equation will be of the form 2x - 5y = k, where k is a constant different from 10.
James sold CD's for $5 each after he paid $15 for equipment and supplies. The equation y=5x-15 shows the profit James will make from selling CD's where x is the number of CD's and y is the profit he makes. If James has 20 CD's to sell, find the domain and range of the function.
Domain: {0, 1, 2, 20}
Y=5(0)-15=-15
Y=5(1)-15=-10
У=5(2)-15=-5
y=5(20)-15=8
Range: {-15, -10, -5,000
Explain how to find the GCF of two polynomials.
Finding the GCF of polynomials involves factoring each polynomial completely and then identifying the common factors. For example, consider 6x² + 12x and 9x³. Factoring gives 6x(x+2) and 9x³. The common factor is 3x.
If 2ˣ = 8 and 3ʸ = 27, what is the value of 2ˣ+ʸ?
First, solve for x and y individually. 2ˣ = 8 implies x = 3 (since 2³ = 8). 3ʸ = 27 implies y = 3 (since 3³ = 27). Therefore, 2ˣ⁺ʸ = 2³⁺³ = 2⁶ = 64.
Expand (2x² - 5)(x² + 3)
This problem involves binomials with squared terms, but the method remains the same.
Two lines are perpendicular. One has the equation 3x + 2y = 7. Write the general form of the equation of the perpendicular line in standard form.
The slope of 3x + 2y = 7 is -3/2. The perpendicular slope is 2/3. The equation of the perpendicular line will be of the form 2x - 3y = k, where k is any constant.