The **range** of a **function** represents all the possible output values it can produce, depending on the input values it receives. For the **functions** given, namely** ( y = 2x ), ( y = 2(5x) ), ( y = 5x + 2 )**, only the **function** **( y = 5x + 2 )** has a **range** of $(2, \infty) $.

This is because as ( x ) approaches **infinity,** the value of ( y ) will grow **indefinitely,** while the lowest value it can take is just above 2 when ( x ) is very close to **zero.**

Understanding the **range** of a **function** helps us visualize how a **function** behaves across the spectrum of real numbers and is particularly essential in fields such as engineering and the physical sciences, where **real-world phenomena** are modeled **mathematically.**

I always find joy in exploring how **equations** manifest in graphs – those seemingly simple lines and curves can tell us so much about the relationships they represent. Let’s go on a journey to decode these **mathematical expressions** together!

## Understanding the Range of Functions

In this section, I’ll explain how to **identify** the **range** of a specific type of **function,** which is crucial for understanding how functions behave on an interval. Let’s dive into how this applies to finding a range that starts at (2) and extends to **infinity.**

### Identifying the Range

To determine the **range** of a **function,** I look at the output values it can produce. The **domain** refers to all possible input values, usually **real numbers**, while the **range** consists of all possible outputs. Let’s take an example of the given **functions:**

- y = 2x
- y = 2(5x)
- y = 5x + 2
- $y = 5^x + 2$

By examining these **functions,** I can see that each one has a different structure, which affects its range. For linear **functions** like (y = 2x) and (y = 2(5x)), their ranges are all **real numbers** because they continue indefinitely in both the positive and negative directions.

However, when I look at the function (y = 5x + 2), if I’m not careful, I could easily overlook that this is still a linear function with an **undefined** range of **infinity** in both directions.

The positive or negative outcome depends on the value of (x), whether it’s positive or negative.

The function that fits the range of $(2, \infty)$ is $y = 5^x + 2$. It’s because this function is not linear; it’s an exponential function where $5^x$ is always **positive**, and since we’re adding (2), it shifts the graph up by (2) units.

This means for any **real number** value of (x), (y) will be greater than (2), approaching infinity but never touching (2), hence the interval $(2, \infty)$.

Let’s tabulate the function alongside its range for clarity:

Function | Range |
---|---|

y = 2x | $(-\infty, \infty)$ |

y = 2(5x) | $(-\infty, \infty)$ |

y = 5x + 2 | $(-\infty, \infty)$ |

$y = 5^x + 2$ | $(2, \infty)$ |

When solving for the **range** mathematically, I also check the **graph**. If I plotted $y = 5^x + 2$, the curve (not a **parabola** or **cubic**) would start at the **point** (0,2) and rise indefinitely as (x) increases.

The **x-axis** represents the **domain** and the (y)-values, given by the vertical axis, clearly show the range starting just above (2) and increasing without bound.

In **algebra**, knowing the range helps to understand the behavior of **functions** and solve problems that have **restrictions** on the output values.

Through learning to interpret these **intervals**, **zero** becomes a pivotal **point** because it can represent an **undefined** state in some functions or the boundary of an **interval** in others.

## Conclusion

In exploring the various **functions,** I’ve found that the **range** of a function tells us the set of possible output values based on the input values it receives. When we talk about the **range** being **$(2, \infty)$**, we’re referring to all real numbers greater than 2, but not including 2 itself. Among the **functions** examined, **$y = 2x$, $y = 2(5x)$, and $y = 5x + 2$**, the one that clearly stands out is** $y = 5x + 2$**.

The **function $y = 5x$** scaled **vertically** by a factor of 2 and then translated **vertically** upward by 2 units, yields** $y = 5x + 2$**.

This translation ensures that the smallest value of **$y$** is just above 2 when **$x$** approaches 0, and as **$x$** increases, **$y$** goes beyond all bounds, approaching infinity. Thus, the **range** of **$y = 5x + 2$** is indeed **$(2, \infty)$**, adhering to the criteria.

My **analysis** confirms that this is the only **function** among the given options that satisfies the condition for the **range**. It is crucial to comprehend that linear **transformations** affect the **range** in predictable ways, which helps in deducing the correct **function** out of the given choices.