For ƒ(x) = 3^x and g(x)= (1/4)^x find each of the following. Round answers to the nearest thousandth as needed. ƒ(-1.68)

Verified step by step guidance

1. Identify the function given: \( f(x) = 3^x \). We need to find \( f(-1.68) \), which means substituting \( x = -1.68 \) into the function.
2. Rewrite the expression for \( f(-1.68) \) as \( 3^{-1.68} \). This means raising 3 to the power of -1.68.
3. Recall that a negative exponent means taking the reciprocal: \( a^{-b} = \frac{1}{a^b} \). So, \( 3^{-1.68} = \frac{1}{3^{1.68}} \).
4. Use a calculator to find the value of \( 3^{1.68} \). This involves using the exponentiation function on your calculator or a computational tool.
5. Finally, take the reciprocal of the result from the previous step to get \( f(-1.68) \), and round your answer to the nearest thousandth as instructed.

Key Concepts

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is a positive constant. They model growth or decay processes and are evaluated by raising the base to the power of the input x. Understanding how to compute values for negative and fractional exponents is essential.

Evaluating Functions at Specific Inputs

Evaluating a function means substituting a given value for the variable and simplifying the expression. For example, to find f(-1.68), replace x with -1.68 and calculate the result, often requiring the use of a calculator for non-integer exponents.

Rounding Decimal Values

Rounding is the process of limiting the number of decimal places to make answers more manageable. Here, answers should be rounded to the nearest thousandth, meaning three digits after the decimal point, which helps in presenting precise yet concise results.