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Evaluating functions is about finding the output for a given input. Think of it like a vending machine where you input a number, and the function gives you an output. For the function \( h(x) = x^{1/4} \), it involves operations with exponents and radicals. Evaluating \( h(81) \) requires substituting \( x \) with \( 81 \). Since \( 81 \) can be expressed as \( 3^4 \), finding \( 81^{1/4} \) simplifies to taking the fourth root of \( 81 \). This simplification process shows that \( (3^4)^{1/4} = 3 \). Therefore, \( h(81) = 3 \). To evaluate a function:

• Substitute the input into the function.
• Perform the necessary calculation.
• Express the result in simplest form.

Understanding the domain and range of a function is vital to grasp its behavior. The domain represents all possible input values that a function can accept. For \( h(x) = x^{1/4} \), the expression is defined for non-negative numbers only. This is because taking the fourth root of a negative number results in a complex number, not a real one. Therefore, the domain is \([0, \, \infty)\), which means all values starting from zero up to positive infinity can be used as inputs.On the other hand, the range of a function is the set of possible output values. For our function, it outputs real non-negative numbers, as the fourth root of zero remains zero, and the fourth root of any positive number also stays positive. Therefore, the range is \([0, \, \infty)\). Knowing the domain and range helps in sketching graphs and understanding the possibilities of a function's outputs.

Exponents and radicals often pair together in functions like \( h(x) = x^{1/4} \), where understanding both is essential for calculations. Exponents indicate repeated multiplication, such as \( 81 = 3^4 \). Radicals, on the other hand, tell us to find the root. The expression \( x^{1/n} \) denotes the \( n \)-th root of \( x \). In our function, \( x^{1/4} \) implies the fourth root of \( x \). Here's the process to simplify expressions involving exponents and radicals:

• Identify the base. For \( 81 \), it is \( 3^4 \).
• Use the rule \( (a^m)^{1/n} = a^{m/n} \) to find the root.
• Simplify to get the result, here \( (3^4)^{1/4} = 3 \).

Mastering these concepts allows you to manipulate and understand complex expressions easily.