f(x)>0
The inequality statement f(x) > 0 represents a mathematical concept in which the function f(x) will have values greater than zero
The inequality statement f(x) > 0 represents a mathematical concept in which the function f(x) will have values greater than zero.
To understand this inequality better, let’s break it down into its key components:
1. f(x): This represents a mathematical function that depends on the variable x. The function can be any mathematical expression involving x, such as f(x) = x^2 + 3x – 2 or f(x) = sin(x).
2. >: This symbol denotes “greater than.” It indicates that the values of f(x) are to be greater than the value on the right-hand side of the inequality.
3. 0: This number represents zero. In our case, we are interested in finding the values of x for which f(x) is greater than zero.
The inequality f(x) > 0 essentially means that we are looking for the values of x for which the function f(x) yields positive values. In other words, we want to determine the part of the function’s graph that lies above the x-axis.
To solve this inequality, you need to analyze the function f(x) and determine the intervals or regions where the function is greater than zero. There are various methods to do this, depending on the complexity of the function:
1. Graphical Method: Plot the graph of the function f(x). Identify the regions on the graph where the function lies above the x-axis (i.e., where f(x) > 0).
2. Algebraic Method: For simpler functions, you can try to solve the inequality algebraically. Start by setting the function f(x) equal to zero and determine the x-values that solve the equation. These points, called critical points, divide the x-axis into intervals. Test a value from each interval in the original inequality to determine if the function is greater than zero in that interval.
Let’s consider an example to illustrate the process:
Example: Solve the inequality f(x) = x^2 – 4x + 3 > 0.
1. Graphical Method: Plotting the function f(x) on a graph, we can see that the graph is a parabola that opens upwards. The parabola intersects the x-axis at x = 1 and x = 3. The part of the graph that lies above the x-axis represents the values of x for which f(x) > 0. Therefore, the solution is x ∈ (1, 3).
2. Algebraic Method: To solve the inequality algebraically, we start by setting f(x) = 0:
x^2 – 4x + 3 = 0
Factoring this quadratic equation, we get:
(x – 1)(x – 3) = 0
Setting each factor equal to zero, we find the critical points:
x – 1 = 0 –> x = 1
x – 3 = 0 –> x = 3
These critical points divide the x-axis into three intervals: (-∞, 1), (1, 3), and (3, ∞). We now test one value from each interval in the original inequality f(x) > 0.
For x < 1, let's consider x = 0: f(0) = (0)^2 - 4(0) + 3 = 3 > 0
For 1 < x < 3, let's consider x = 2:
f(2) = (2)^2 - 4(2) + 3 = -1 < 0
For x > 3, let’s consider x = 4:
f(4) = (4)^2 – 4(4) + 3 = 3 > 0
Based on these test values, we find that f(x) > 0 for x ∈ (-∞, 1) and x ∈ (3, ∞). Therefore, the solution to the inequality is x ∈ (-∞, 1) ∪ (3, ∞).
Remember that the specific method used to solve these types of inequalities may vary depending on the complexity of the function.
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