Question

True or False. The \(x\) -intercepts of the graph of a function \(y=f(x)\) are the real solutions of the equation \(f(x)=0 .\)

Short answer

True

Step-by-step solution

- Understand the Definition of x-intercepts

The x-intercepts of the graph of a function are the points where the graph crosses the x-axis. At these points, the value of y is zero.

- Formulate the Condition

For the graph to intersect the x-axis, the y-coordinate must be zero. Therefore, we need to find the values of x for which y equals zero. This translates to solving the equation: f(x) = 0

- Connect the Formulations

The above analysis shows that the real solutions to the equation f(x) = 0 are precisely the x-coordinates of the points where the graph of the function intersects the x-axis.

- Conclusion

Given that the x-intercepts of the graph of a function are determined by setting y to zero and solving for x, it is true that the x-intercepts are the real solutions of the equation f(x)=0.

Key concepts

Graph of a Function

The graph of a function visually represents the relationship between the input and the output of the function. Each point on the graph corresponds to a pair \((x, y)\), where \(y = f(x)\). When sketching or analyzing a graph, look for key features like intercepts, slopes, and the shape of the graph.

The x-intercepts and y-intercepts are particularly important. The x-intercepts are points where the function crosses the x-axis, meaning the value of \(y\) is zero at these points. Conversely, the y-intercept is where the function crosses the y-axis, which happens when \(x = 0\).

By examining the graph, you can understand the behavior of the function and how it changes across different values of \(x\). This visual representation makes it easier to identify solutions, trends, and anomalies within the function.

Real Solutions

In algebra, real solutions refer to the values of \(x\) that satisfy the equation within the real number system. This means these solutions are not imaginary or complex numbers.

When solving an equation like \(f(x) = 0\), you are seeking the x-values that make the function equal to zero. These real solutions correspond to the x-intercepts on the graph of the function.

Finding real solutions might involve different algebraic techniques such as factoring, using the quadratic formula, or employing numerical methods for more complex functions. Knowing these real solutions helps in understanding the function's behavior and in predicting its values at certain points.

f(x) = 0

The equation \(f(x) = 0\) is central in finding the x-intercepts of the graph of a function. It states that the function's output is zero, and you are required to determine which input values (x-values) cause this to happen.

Solving \(f(x) = 0\) involves setting the function equal to zero and finding the x-values that fulfill this condition. For example, if \(f(x) = x^2 - 4\), you solve \(x^2 - 4 = 0\) to get \(x = 2\) and \(x = -2\). These x-values are the points where the function crosses the x-axis.

This relationship shows that the x-intercepts are indeed the real solutions of the equation \(f(x) = 0\). Understanding this connection is crucial for graphing functions and solving algebraic equations effectively.