Is y = 2x + 5 a function?

Understand the Problem

The question is asking whether the equation y = 2x + 5 represents a function. To determine if it is a function, we need to see if each input x corresponds to exactly one output y. In this case, it is a linear equation, which represents a function as every input will yield a unique output.

Answer

Yes, the equation $y = 2x + 5$ represents a function.

Steps to Solve

Identifying the Equation Type

The equation given is $y = 2x + 5$, which is in the form of $y = mx + b$, where $m$ is the slope (2) and $b$ is the y-intercept (5). This form indicates that it's a linear equation.

Understanding Functions

A function is a relation where each input (x) has exactly one output (y). In a linear equation like this, for any value of $x$, you can calculate a unique value of $y$.

Checking Input and Output

To confirm that this equation represents a function, consider plugging in different values for $x$.

For instance:

If $x = 1$, then $y = 2(1) + 5 = 7$
If $x = 2$, then $y = 2(2) + 5 = 9$
If $x = 3$, then $y = 2(3) + 5 = 11$

Each input gives a unique output, which confirms it maintains the function property.

Conclusion about the Function

Since every input $x$ results in a unique output $y$, we conclude that $y = 2x + 5$ is indeed a function.

Yes, the equation $y = 2x + 5$ represents a function.

More Information

This equation is a classic example of a linear function, which is characterized by a constant rate of change. Linear functions are fundamental in algebra and are often used to model relationships between two variables.

Tips

Confusing linear equations with non-linear ones, such as quadratic equations, which can produce multiple outputs for a single input.
Forgetting that a vertical line test can help determine if a relation is a function. If any vertical line intersects the graph of the equation more than once, it is not a function.

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