7. Find The Domain And Range Of The Following Functions.a) \[$ F(x) = 1 + 8x - 2x^2 \$\]b) \[$ F(x) = \frac{1}{x^2 - 5x + 6} \$\]c) \[$ F(x) = \sqrt{x^2 - 6x + 8} \$\]d) \[$ F(x) = \begin{cases} 3x + 4, & -1 \leq X \

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In mathematics, functions are used to describe the relationship between variables. Understanding the domain and range of a function is crucial in various mathematical and real-world applications. The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce.

In this article, we will explore the domain and range of four different functions: a quadratic function, a rational function, a square root function, and a piecewise function.

a) Domain and Range of the Quadratic Function

The quadratic function is given by:

f(x)=1+8x−2x2 f(x) = 1 + 8x - 2x^2

To find the domain of this function, we need to determine the values of x for which the function is defined. Since the function is a polynomial, it is defined for all real numbers. Therefore, the domain of the function is all real numbers, denoted by (−∞,∞)(-\infty, \infty).

To find the range of the function, we need to determine the set of all possible output values. We can do this by finding the vertex of the parabola and determining the direction of the opening. The vertex of the parabola is given by:

x=−b2a=−82(−2)=2 x = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2

Substituting this value into the function, we get:

f(2)=1+8(2)−2(2)2=1+16−8=9 f(2) = 1 + 8(2) - 2(2)^2 = 1 + 16 - 8 = 9

Since the parabola opens downward, the range of the function is all real numbers less than or equal to 9, denoted by (−∞,9](-\infty, 9].

b) Domain and Range of the Rational Function

The rational function is given by:

f(x)=1x2−5x+6 f(x) = \frac{1}{x^2 - 5x + 6}

To find the domain of this function, we need to determine the values of x for which the function is defined. Since the function is a rational function, it is undefined when the denominator is equal to zero. We can find the values of x that make the denominator equal to zero by solving the quadratic equation:

x2−5x+6=0 x^2 - 5x + 6 = 0

Factoring the quadratic equation, we get:

(x−2)(x−3)=0 (x - 2)(x - 3) = 0

Solving for x, we get:

x=2 or x=3 x = 2 \text{ or } x = 3

Therefore, the domain of the function is all real numbers except 2 and 3, denoted by (−∞,2)∪(2,3)∪(3,∞)(-\infty, 2) \cup (2, 3) \cup (3, \infty).

To find the range of the function, we need to determine the set of all possible output values. Since the function is a rational function, the range is all real numbers except zero.

c) Domain and Range of the Square Root Function

The square root function is given by:

f(x)=x2−6x+8 f(x) = \sqrt{x^2 - 6x + 8}

To find the domain of this function, we need to determine the values of x for which the function is defined. Since the function is a square root function, it is defined only when the expression inside the square root is non-negative. We can find the values of x that make the expression inside the square root non-negative by solving the quadratic inequality:

x2−6x+8≥0 x^2 - 6x + 8 \geq 0

Factoring the quadratic expression, we get:

(x−2)(x−4)≥0 (x - 2)(x - 4) \geq 0

Solving for x, we get:

x≤2 or x≥4 x \leq 2 \text{ or } x \geq 4

Therefore, the domain of the function is all real numbers less than or equal to 2 and all real numbers greater than or equal to 4, denoted by (−∞,2]∪[4,∞)(-\infty, 2] \cup [4, \infty).

To find the range of the function, we need to determine the set of all possible output values. Since the function is a square root function, the range is all non-negative real numbers.

d) Domain and Range of the Piecewise Function

The piecewise function is given by:

f(x)={3x+4,−1≤x≤22x−1,x>2 f(x) = \begin{cases} 3x + 4, & -1 \leq x \leq 2 \\ 2x - 1, & x > 2 \end{cases}

To find the domain of this function, we need to determine the values of x for which the function is defined. Since the function is a piecewise function, it is defined for all real numbers except the values that make the expressions inside the piecewise function undefined. In this case, the expressions inside the piecewise function are polynomials, so they are defined for all real numbers.

Therefore, the domain of the function is all real numbers, denoted by (−∞,∞)(-\infty, \infty).

To find the range of the function, we need to determine the set of all possible output values. We can do this by finding the values of the function at the endpoints of the intervals and determining the direction of the function.

At x = -1, we have:

f(−1)=3(−1)+4=−3+4=1 f(-1) = 3(-1) + 4 = -3 + 4 = 1

At x = 2, we have:

f(2)=3(2)+4=6+4=10 f(2) = 3(2) + 4 = 6 + 4 = 10

Since the function is increasing on the interval (−1,2)(-1, 2) and decreasing on the interval (2,∞)(2, \infty), the range of the function is all real numbers greater than or equal to 1 and all real numbers less than or equal to 10, denoted by [1,∞)∪(−∞,10][1, \infty) \cup (-\infty, 10].

In the previous article, we explored the domain and range of four different functions: a quadratic function, a rational function, a square root function, and a piecewise function. In this article, we will answer some frequently asked questions about the domain and range of functions.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values (y-values) that the function can produce.

Q: How do I find the domain of a function?

A: To find the domain of a function, you need to determine the values of x for which the function is defined. This can be done by:

  • Checking if the function is a polynomial, in which case the domain is all real numbers.
  • Checking if the function is a rational function, in which case the domain is all real numbers except the values that make the denominator equal to zero.
  • Checking if the function is a square root function, in which case the domain is all real numbers that make the expression inside the square root non-negative.
  • Checking if the function is a piecewise function, in which case the domain is all real numbers except the values that make the expressions inside the piecewise function undefined.

Q: How do I find the range of a function?

A: To find the range of a function, you need to determine the set of all possible output values. This can be done by:

  • Finding the values of the function at the endpoints of the intervals.
  • Determining the direction of the function on each interval.
  • Using the domain of the function to determine the range.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can a function have a domain that is not a set of real numbers?

A: Yes, a function can have a domain that is not a set of real numbers. For example, a function that takes complex numbers as input can have a domain that is a set of complex numbers.

Q: Can a function have a range that is not a set of real numbers?

A: Yes, a function can have a range that is not a set of real numbers. For example, a function that takes complex numbers as output can have a range that is a set of complex numbers.

Q: How do I graph a function with a domain and range that are not sets of real numbers?

A: To graph a function with a domain and range that are not sets of real numbers, you can use complex numbers to represent the input and output values. You can also use other mathematical tools, such as matrices and vectors, to represent the function.

Q: What are some common mistakes to avoid when finding the domain and range of a function?

A: Some common mistakes to avoid when finding the domain and range of a function include:

  • Assuming that the domain and range are always sets of real numbers.
  • Failing to check for values that make the denominator equal to zero in rational functions.
  • Failing to check for values that make the expression inside the square root non-negative in square root functions.
  • Failing to determine the direction of the function on each interval.

By understanding the domain and range of functions, you can determine the set of all possible input values and output values, which is crucial in various mathematical and real-world applications.