Let be a positive constant. The graph of the equation below has no -intercepts.
Which choice could be the value of ?
For rational equations involving exponentials, first treat the repeated exponential expression as one quantity and factor. Then use the original denominator to identify excluded inputs. A canceled factor can create a missing point, so a numerator zero is an -intercept only if the original denominator is not zero at that same input.
Hints
Use a substitution
Because both and appear, let and factor the resulting quadratic in .
Do not cancel without tracking the domain
A value that makes the denominator zero is not part of the graph, even if the same factor appears in the numerator.
Examine the remaining factor
After factoring, determine when is excluded by the denominator.
Desmos Guide
Use algebra first
Factoring is the fastest method because it shows both the possible zeros and the excluded value from the denominator.
Enter the function
Enter
and create a slider for .
Check the candidate values
Set to each answer choice and add a table with . For , , and , the table gives . For , the table entry at is undefined because both the numerator and denominator are zero there, confirming that the apparent intercept is excluded.
Step-by-step Explanation
Rewrite the numerator
Let . Then the numerator becomes
Thus, the equation can be written as
Account for the restricted value
The expression is undefined when , or when . Therefore, a zero from the factor cannot be an -intercept because that value of is not in the graph's domain.
Check the remaining possible intercept
The other numerator factor is zero when , which occurs at . This is an -intercept unless the denominator is also zero there. At , the denominator is , so it is zero only when .
When , the only possible zero is excluded from the domain, leaving no -intercepts.