Understanding the Range of the Function: f(x) 2x^2 - 4x 3

When dealing with mathematical functions, the range is an important concept to understand. The range of a function is the set of all possible output values (y-values) that the function can produce. In the context of quadratic functions, such as ( f(x) 2x^2 - 4x 3 ), determining the range involves analyzing the function's vertex and its maximum or minimum value.

Domain and Range in Different Contexts

In this case, we are primarily interested in the range of ( f(x) 2x^2 - 4x 3 ). However, it's important to note that the range can vary depending on the domain, or the set of input values (x-values) for which the function is defined. Here are some scenarios:

If the domain is all real numbers ((mathbb{R})): In this case, the function ( f(x) 2x^2 - 4x 3 ) is a parabola that opens upwards (since the coefficient of ( x^2 ) is positive). The range of such a function is all real numbers greater than or equal to the minimum value of the function. If the domain is the set of complex numbers ((mathbb{C})): The range would then include all complex numbers, as a quadratic function with complex coefficients and real or complex roots can produce any complex output. If the domain is the finite set (mathbb{Z} / 5mathbb{Z}): Here, the range would be the set of function values for the specific input values in this finite field, which in this case is ({1, 3, 4}).

Finding the Minimum Value

For the purpose of finding the range of ( f(x) 2x^2 - 4x 3 ), let's consider the domain to be all real numbers. To find the minimum value, we need to analyze the first and second derivatives of the function:

First Derivative: Set the first derivative to zero to find the critical points.

f'(x) 4x - 4
4x - 4 0
4x 4
x 1

Second Derivative: To determine whether this critical point is a minimum or maximum, we check the second derivative.

f''(x) 4
f''(1) 4 0

Since the second derivative is positive, the function has a minimum at (x 1).

Evaluating the Minimum Value: Substitute (x 1) back into the original function to find the minimum value.

f(1) 2(1)^2 - 4(1) 3 2 - 4 3 1

The minimum value of the function is 1.

Vertex Form and Range

Another way to understand the range is by expressing the function in vertex form. This form helps us directly identify the vertex of the parabola, which is the point where the minimum value occurs:

Start with the given function:

f(x) 2x^2 - 4x 3

Complete the square to rewrite the function in vertex form:

f(x) 2(x^2 - 2x) 3
f(x) 2(x^2 - 2x 1 - 1) 3
f(x) 2((x - 1)^2 - 1) 3
f(x) 2(x - 1)^2 - 2 3
f(x) 2(x - 1)^2 1

The vertex form is ( f(x) 2(x - 1)^2 1 ), indicating that the vertex is at ((1, 1)).

Since the parabola opens upwards ((a 2 > 0)), the range of the function is all real numbers greater than or equal to the minimum value, which is 1.

Conclusion

In summary, the range of the function ( f(x) 2x^2 - 4x 3 ) is all real numbers greater than or equal to 1, as it represents a parabola that opens upwards with a minimum value of 1.