Question Video: Evaluating an Exponential Function at Given Values Mathematics

Video Transcript

Suppose that 𝑓 of 𝑥 is equal to 0.3 multiplied by four to the power of 𝑥. Evaluate 𝑓 of zero, evaluate 𝑓 of two, evaluate 𝑓 of 0.5, and evaluate 𝑓 of negative one-half.

In this question, we’re given a function 𝑓 of 𝑥, which we can see is an exponential function. And we’re asked to evaluate this exponential function at four different values. To do this, we can recall to evaluate a function at a value, we need to substitute the value of the variable into the function. For example, to evaluate 𝑓 at zero, we substitute 𝑥 is equal to zero into the function. We get 𝑓 of zero is 0.3 multiplied by four to the zeroth power. We can then evaluate this expression by recalling any nonzero number raised to the zeroth power is equal to one. Therefore, four to the zeroth power is equal to one. So we get 0.3 multiplied by one, which is equal to 0.3. Hence, 𝑓 evaluated at zero is equal to 0.3.

We can follow this process to evaluate 𝑓 at two. This time, we substitute 𝑥 is equal to two into the function. We get 𝑓 evaluated at two is 0.3 multiplied by four squared. And we can evaluate this by recalling squaring a number means we multiply it by itself. So four squared is equal to four times four, which is equal to 16. And we can then evaluate this expression. 0.3 multiplied by 16 is 4.8. So 𝑓 evaluated at two is equal to 4.8.

We now want to follow the same process to evaluate 𝑓 at 0.5. We substitute 𝑥 is equal to 0.5 into the function to get 0.3 multiplied by four to the power of 0.5. And to evaluate this, we’re going to need to rewrite four to the power of 0.5 as four to the power of one-half. We can then evaluate this by recalling when we raise a number to an exponent of one-half, we mean the square root of that number. So four to the power of one-half is the square root of four, which we know is two. So we get 0.3 multiplied by two. And we can evaluate this. 0.3 multiplied by two is equal to 0.6. Therefore, 𝑓 evaluated at 0.5 is equal to 0.6.

We now need to follow this process one more time to evaluate 𝑓 at negative one-half. We substitute 𝑥 is equal to negative one-half into our function to get 0.3 multiplied by four to the power of negative one-half. And we can simplify this expression by recalling one of the laws of exponents. 𝑏 to the power of negative 𝑛 is equal to one divided by 𝑏 to the 𝑛th power. Therefore, we can rewrite four to the power of negative one-half as one divided by four to the power of one-half.

But remember, four to the power of one-half is the square root of four, which we know is equal to two. Therefore, we can simplify this expression to be 0.3 multiplied by one-half, which we can then evaluate to give us 0.15. Therefore, we were able to show if 𝑓 of 𝑥 is equal to 0.3 times four to the power of 𝑥, then 𝑓 evaluated at zero is equal to 0.3. 𝑓 evaluated at two is equal to 4.8. 𝑓 evaluated at 0.5 is equal to 0.6. And 𝑓 evaluated at negative one-half is equal to 0.15.