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.
