{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.




a. What is the domain of f (in terms of a)?

Verified step by step guidance

1. To determine the domain of the function \( f(x) = (a + x)^x \), we need to identify the values of \( x \) for which the expression is defined.
2. The expression \( (a + x)^x \) is defined when \( a + x > 0 \) because the base of the power must be positive for real number exponents.
3. Solve the inequality \( a + x > 0 \) to find the values of \( x \). This gives \( x > -a \).
4. Therefore, the domain of \( f(x) \) in terms of \( a \) is all real numbers \( x \) such that \( x > -a \).
5. In interval notation, the domain is \( (-a, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function f(x) = (a + x)ˣ, where a > 0, the domain is determined by the values of x that keep the expression a + x positive, as the base of an exponent must be positive for real-valued outputs.

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = bˣ, where b is a positive constant. In the case of f(x) = (a + x)ˣ, the function exhibits superexponential growth, meaning it grows faster than any polynomial function as x increases. Understanding the behavior of exponential functions is crucial for analyzing their domains and ranges.

Inequalities

Inequalities are mathematical statements that compare two expressions, indicating whether one is greater than, less than, or equal to the other. In determining the domain of f(x) = (a + x)ˣ, we need to solve the inequality a + x > 0, which leads to x > -a. This understanding of inequalities is essential for identifying valid input values for the function.