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

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!