Question

The domain of the defination of the function f(x) given by the euation an+a₄=a is____(a>1).

• A. 0<x<1
• B. -∞<x<0
• C. 0<x<2
• D. -∞<x<1

Answer 1

The Correct Option is D

We need to determine the domain of the function \( f(x) \) defined by the equation \( a^x + a^{f(x)} = a \), where \( a > 1 \).

1. Interpret the Equation:
The given equation is \( a^x + a^{f(x)} = a \). We need to find the values of \( x \) for which \( f(x) \) is defined, i.e., the domain of \( f(x) \). This means we must solve for \( f(x) \) in terms of \( x \) and determine the values of \( x \) that produce a real \( f(x) \).

2. Solve for \( f(x) \):
Start with the equation:

\( a^x + a^{f(x)} = a \)
Isolate \( a^{f(x)} \):

\( a^{f(x)} = a - a^x \)
For \( a^{f(x)} \) to be defined and real, the right-hand side must be positive (since \( a^{f(x)} > 0 \) for all real \( f(x) \), as \( a > 1 \)):

\( a - a^x > 0 \)
\( a > a^x \)
Since \( a > 1 \), divide both sides by \( a \):

\( 1 > \frac{a^x}{a} = a^{x-1} \)
Thus:

\( a^{x-1} < 1 \)
Since \( a > 1 \), \( a^{x-1} < 1 \) implies the exponent must be negative:

\( x - 1 < 0 \)
\( x < 1 \)

3. Express \( f(x) \):
Now, take the natural logarithm on both sides of \( a^{f(x)} = a - a^x \):

\( f(x) \ln a = \ln (a - a^x) \)
\( f(x) = \frac{\ln (a - a^x)}{\ln a} \)
For \( f(x) \) to be real, the argument of the logarithm must be positive, which we already established as \( a - a^x > 0 \), or \( x < 1 \). Additionally, \( \ln a > 0 \) since \( a > 1 \), so the denominator is positive and well-defined.

4. Check Boundary Conditions:
- As \( x \to 1^- \), \( a^x \to a^1 = a \), so \( a - a^x \to a - a = 0 \), making \( \ln (a - a^x) \to -\infty \), and thus \( f(x) \to -\infty \), which is still real.
- For \( x < 1 \), \( a - a^x > 0 \), so \( \ln (a - a^x) \) is real, and \( f(x) \) is defined.
- If \( x \geq 1 \), then \( a - a^x \leq 0 \), making \( \ln (a - a^x) \) undefined (since the logarithm of a non-positive number is not real).

5. Determine the Domain:
The condition \( x < 1 \) ensures \( a - a^x > 0 \), allowing \( f(x) \) to be defined. There are no other restrictions, as \( a > 1 \) ensures all terms are well-behaved for \( x < 1 \). Thus, the domain of \( f(x) \) is all real numbers less than 1.

Final Answer:
The domain of the function \( f(x) \) defined by the equation \( a^x + a^{f(x)} = a \) (where \( a > 1 \)) is: \( (-\infty, 1) \)