1. Provide two examples of a function: (1) as a table of values and (2) as a graph. Explain the definition of a function. Be sure to include mathematical vocabulary in your response.
2. In a paragraph, explain what the domain and range is for the function f(x) = x2 + 4x - 21.
3. An equation, f, has a domain of all whole numbers and has a range of all real numbers. Does the equation represent a function? Explain why or why not.
4. What is the range of the function g(x) = 3x2 - 6x + 3 when the domain is defined as the set of integers, x, such that 0<=x<=4? Show all work.
5. Write the recursive formula and the explicit formula for the sequence {-15,-7,1,9, 17,...}. Show all work.
6. In a paragraph, write which formula, recursive or explicit, is easiest for you to use when finding the 10th term in a sequence. Explain your reasoning in complete sentences.
1. A function is a mathematical relationship between two sets, called the domain and the range, such that each element in the domain is associated with exactly one element in the range. Here are two examples of functions:
Example 1 (as a table of values):
Let's consider the function f(x) = 2x + 1, where x is the input variable and f(x) is the output. We can create a table of values:
|x | f(x) |
|---|------|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
In this table, each value in the domain (x) corresponds to a unique value in the range (f(x)), satisfying the definition of a function.
Example 2 (as a graph):
Consider the function g(x) = x^2, which is a quadratic function. We can graph this function on a coordinate plane. The x-axis represents the input variable, and the y-axis represents the output:
^
| |
| |
| .....|
--------|------------------------------------->
| |
| |
| |
In this graph, each vertical line intersects with the curve at exactly one point, indicating that for each value of x, there is a unique corresponding value of y. Thus, it is a function.
2. The given function f(x) = x^2 + 4x - 21 has a quadratic expression. The domain refers to all possible input values, and the range refers to all possible output values. In this case, since it is a quadratic equation, it is a parabola facing upward, which means it has a minimum value. To determine the domain and range, we consider the shape of the parabola.
The domain is all real numbers because there are no restrictions on the input variable, x.
To find the range, we need to determine the minimum point of the parabola. The x-coordinate of the minimum point can be found using the formula x = -b/2a, where a is the coefficient of x^2 (1 in this case) and b is the coefficient of x (4 in this case). Plugging in the values, we get x = -4/(2*1) = -2.
Substituting this value back into the function to find the corresponding y-coordinate, we get f(-2) = (-2)^2 + 4(-2) - 21 = 4 - 8 - 21 = -25.
Therefore, the range of the function is (-∞, -25), as the function can take any value less than -25.
3. If an equation has a domain consisting of all whole numbers and a range consisting of all real numbers, it does represent a function. The definition of a function states that for each element in the domain, there must be exactly one corresponding element in the range. In this case, since every whole number has a corresponding real number, the equation satisfies this requirement and thus represents a function.
4. To find the range of the function g(x) = 3x^2 - 6x + 3 when the domain is defined as the set of integers such that 0<=x<=4, we evaluate the function for each integer value in the given domain and determine the range.
When x = 0: g(0) = 3(0)^2 - 6(0) + 3 = 3
When x = 1: g(1) = 3(1)^2 - 6(1) + 3 = 0
When x = 2: g(2) = 3(2)^2 - 6(2) + 3 = 3
When x = 3: g(3) = 3(3)^2 - 6(3) + 3 = 12
When x = 4: g(4) = 3(4)^2 - 6(4) + 3 = 27
The range of the given function, for the given domain, is {0, 3, 12, 27}.
5. To find the recursive and explicit formulas for the sequence {-15, -7, 1, 9, 17, ...}, we need to observe the pattern. Here, each term is obtained by adding 8 to the previous term.
Recursive formula: We can express the sequence recursively by defining a starting term and a recursive relationship. Let's call the first term a₁. The recursive formula is:
a₁ = -15
aₙ = aₙ₋₁ + 8, for n > 1
Explicit formula: An explicit formula gives a direct expression for finding the nth term. To find the explicit formula, we need to determine the common difference between terms. In this case, the common difference is 8. Let's denote the term number as n.
a(n) = a₁ + (n-1) * d
= -15 + (n-1) * 8
= -15 + 8n - 8
= 8n - 23
Therefore, the recursive formula is aₙ = aₙ₋₁ + 8, and the explicit formula is a(n) = 8n - 23.
6. Personally, I find the explicit formula easier to use when finding the 10th term in a sequence. The explicit formula directly gives the nth term without having to calculate each term before it. This saves time and effort, especially for longer sequences or higher term numbers. Additionally, once you have the explicit formula, plugging in the desired term number into the formula is relatively straightforward and efficient. Overall, the explicit formula is more convenient for quickly finding specific terms in a sequence.