$$ y = \frac { 10 ^ { x } + 10 ^ { - x } } { 2 } $$

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The given function is

$$
y = \frac{10^x + 10^{-x}}{2}
$$

This function can be analyzed for various features such as its domain, range, and behavior. Let's go through these aspects step by step.

### Step 1: Domain
The domain of the function is the set of all possible values of $$x$$ for which the function is defined. Since $$10^x$$ and $$10^{-x}$$ are defined for all real numbers, the domain of $$y$$ is:

$$
\text{Domain: } (-\infty, \infty)
$$

### Step 2: Range
To find the range, we can analyze the behavior of the function. The expression $$10^x + 10^{-x}$$ is always positive for all real $$x$$.

Let's denote $$a = 10^x$$. Then, we can rewrite the function as:

$$
y = \frac{a + \frac{1}{a}}{2}
$$

The expression $$a + \frac{1}{a}$$ reaches its minimum value when $$a = 1$$ (by the AM-GM inequality), which gives:

$$
a + \frac{1}{a} \geq 2
$$

Thus,

$$
y \geq \frac{2}{2} = 1
$$

As $$x$$ approaches $$\infty$$ or $$-\infty$$, $$y$$ approaches $$\infty$$. Therefore, the range of $$y$$ is:

$$
\text{Range: } [1, \infty)
$$

### Step 3: Behavior
- As $$x \to \infty$$, $$y \to \infty$$.
- As $$x \to -\infty$$, $$y \to \infty$$ as well.

### Step 4: Symmetry
The function is even because:

$$
y(-x) = \frac{10^{-x} + 10^{x}}{2} = y(x)
$$

### Conclusion
The function $$y = \frac{10^x + 10^{-x}}{2}$$ has the following characteristics:
- **Domain:** $$(- \infty, \infty)$$
- **Range:** $$[1, \infty)$$
- **Behavior:** Approaches $$\infty$$ as $$x$$ approaches $$\pm \infty$$
- **Symmetry:** Even function

If you need further analysis or specific calculations, please let me know!
The function $$ y = \frac{10^x + 10^{-x}}{2} $$ is an even function with a domain of all real numbers. Its range is from 1 to infinity, and it increases without bound as $$ x $$ approaches both positive and negative infinity.

\( y = \frac { 10 ^ { x } + 10 ^ { - x } } { 2 } \)

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