Use the function below to find $f(-2)$.
$f(x)=5^x$
A. $\frac{1}{25}$
B. -25
C. -10
D. $\frac{1}{10}
Answer (2)
Substitute x = − 2 into the function f ( x ) = 5 x .
Apply the negative exponent rule: 5 − 2 = 5 2 1 .
Calculate 5 2 = 25 .
The final answer is 25 1 .
Explanation
Understanding the problem We are given the function f ( x ) = 5 x and asked to find the value of f ( − 2 ) . This means we need to substitute − 2 for x in the function and evaluate the expression.
Substituting x = -2 To find f ( − 2 ) , we substitute x = − 2 into the function: f ( − 2 ) = 5 − 2 Recall that a negative exponent means we take the reciprocal of the base raised to the positive exponent: a − n = a n 1 .
Applying the negative exponent rule Applying this rule, we have: 5 − 2 = 5 2 1 Now we need to calculate 5 2 , which means 5 multiplied by itself.
Calculating 5 squared Calculating 5 2 , we get: 5 2 = 5 × 5 = 25 Therefore, f ( − 2 ) = 25 1 This matches option A.
Final Answer Thus, the value of f ( − 2 ) is 25 1 .
Examples
Exponential functions like f ( x ) = 5 x are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a bacterial population triples every hour, the population size can be modeled by an exponential function. Similarly, the decay of a radioactive substance can be modeled using an exponential function with a negative exponent. Understanding how to evaluate exponential functions for different values of x is crucial in these applications.
The value of the function f ( − 2 ) for f ( x ) = 5 x is calculated as f ( − 2 ) = 5 − 2 = 25 1 . Therefore, the correct answer is option A. This demonstrates how negative exponents can be evaluated by taking the reciprocal of the base raised to the corresponding positive exponent.
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