Question

what is the positive interval of the equation g(x) = -2^(x) + 4

Answer 1

To find the positive interval of the function \( g(x) = -2^x + 4 \), we need to determine where the function is positive, i.e., where \( g(x) > 0 \).

We can set up the inequality:

\[
-2^x + 4 > 0
\]

Rearranging this gives:

\[
4 > 2^x
\]

or

\[
2^x < 4
\]

Since \( 4 \) can be expressed as a power of \( 2 \), specifically \( 4 = 2^2 \), we can rewrite the inequality as:

\[
2^x < 2^2
\]

Since the base \( 2 \) is positive, we can compare the exponents:

\[
x < 2
\]

Thus, the solution to the inequality is \( x < 2 \).

Now, since we are looking for the interval in positive \( x \), we consider the interval:

\[
(0, 2)
\]

So, the positive interval where \( g(x) > 0 \) is:

\[
(0, 2)
\]