To solve the given equation for [tex]\( k \)[/tex], we start with the equation provided:
[tex]\[
\frac{1}{h(x)} = 2^{kx}
\][/tex]
To isolate [tex]\( k \)[/tex], we need to carefully manipulate the equation. Follow these steps:
1. Take the reciprocal of both sides: This helps in simplifying the equation. The reciprocal of [tex]\( \frac{1}{h(x)} \)[/tex] is [tex]\( h(x) \)[/tex] and the reciprocal of [tex]\( 2^{kx} \)[/tex] is [tex]\( 2^{-kx} \)[/tex]:
[tex]\[
h(x) = 2^{-kx}
\][/tex]
2. Identify the relationship: To find [tex]\( k \)[/tex], we should understand whether there is specific information about [tex]\( h(x) \)[/tex] and how it changes with respect to [tex]\( x \)[/tex]. Unfortunately, with the given information, it's not clear what [tex]\( h(x) \)[/tex] specifically is, but let's assume for a minute that we can work with this general form.
3. Cross-multiplication: If we need to confirm explicitly for [tex]\( k \)[/tex] involving given terms, we recognize standard forms. For example:
Since:
[tex]\[
\frac{1}{h(x)} = 2^{kx}
\][/tex]
When simplified:
[tex]\[
h(x) = 2^{-kx}
\][/tex]
4. Solve for [tex]\( k \)[/tex]: If more context were provided, such as specific values or additional context for [tex]\( h(x) \)[/tex], we could isolate [tex]\( k \)[/tex]. But given the typical nature of such problems and assuming standard simplification, we would normally determine [tex]\( k \)[/tex] once specifics are shared. However, as details are not explicitly specified here, we conclude our steps.
Given the problem as written and without specific values for [tex]\( h(x) \)[/tex] or [tex]\( x \)[/tex], we confidently state that the value of [tex]\( k \)[/tex] in context of provided information would seemingly be indeterminate or requires more context for exact numerical evaluation.
Thus, the value of [tex]\( k \)[/tex]:
[tex]\[
\boxed{None}
\][/tex]
This concludes that with provided outline for interpretations, more specific details are needed or the presented equation in the form implies indeterminacy at the specific phrasing setup present.
