Question

Use the properties of exponents to rewrite the function f(x)=351−x+7
in the form f(x)=a(b)x−h+k
f(x)=3(_)^x-_ +7

Answer 1

To rewrite the function \( f(x) = 351 - x + 7 \) in the form of \( f(x) = a(b)^{x-h} + k \), we first need to simplify the given function.

Starting with your function:

\[
f(x) = 351 - x + 7
\]

Combine the constants:

\[
f(x) = 358 - x
\]

Next, we want to express the function in a form involving exponents. We can extract \( -x \) and rewrite the linear term in terms of an exponential function:

\[
f(x) = 358 - x = 358 - (e^{-\ln(10)} \cdot x)
\]

However, this does not fit directly into the \( a(b)^{x-h} + k \) format you're looking for. Instead, let’s isolate \( x \) in an exponential format.

We can rewrite \( f(x) \) in a more generalized manner:

\[
f(x) = 358 - 1 \cdot e^{\ln(10)} \cdot x
\]

To align with the desired format, define:

- Let \( a = 358 \) (the adjusted vertical shift),
- Let \( b = e^{\ln(10)} = 10 \) (a base value in terms of an exponential function),
- Let \( h = 0 \) (since there is no horizontal shift),
- Let \( k = 0 \) (which fits as part of the writing structure).

This leads us to rewrite the function in the required format:

\[
f(x) = 3(10)^{-x} + 7
\]

However, since we want it in specific terms as you outlined:

\[
f(x) = 3(10)^{-x} + 7
\]

This means we have:

\[
f(x) = 3(10)^{-x} + 7
\]

So the completed function can be summarized as:

\[
f(x) = 3(10)^{x - 0} + 7
\]

In the end, fill in the gaps:

\[
f(x) = 3(10)^{x - 0} + 7
\]

Thus:

- \( a = 3 \)
- \( b = 10 \)
- \( h = 0 \)
- \( k = 7 \)

So the final function is expressed as:

\[
f(x) = 3(10)^{x} + 7
\]