Let's analyze the given equation in the image:

$\left( \frac{1}{3} \right)^{|x|} = x^2 - a$

We are asked to find the range of $a$ such that the equation has two solutions.

Step-by-step approach:

1. Understand the left-hand side: $\left( \frac{1}{3} \right)^{|x|}$ The expression $\left( \frac{1}{3} \right)^{|x|}$ is an exponential function with a base less than 1. This implies that as $|x|$ increases, $\left( \frac{1}{3} \right)^{|x|}$ decreases. Therefore, it is a decreasing function, and it is always positive.

2. Right-hand side: $x^2 - a$ This is a quadratic expression in $x$, shifted down by $a$. For different values of $a$, the parabola will shift vertically, and we need to find when this expression has two solutions.

3. Determine the conditions for two solutions:

   • The equation will have two solutions if the line $\left( \frac{1}{3} \right)^{|x|}$ intersects the curve $x^2 - a$ at two points. Since $\left( \frac{1}{3} \right)^{|x|}$ is a decreasing function and $x^2 - a$ is a parabola opening upwards, for two intersections, the value of $a$ must lie within a certain range.

4. Behavior at extreme values:

   • As $|x| \to \infty$, $\left( \frac{1}{3} \right)^{|x|} \to 0$. Hence, for two solutions, the parabola must be such that it shifts down enough to intersect the exponential curve twice. This implies that $a$ must take values where the quadratic has two distinct intersections with the curve.

Solution:

The range of $a$ that satisfies the condition for two solutions is $(-1, \infty)$. This can be seen from the interplay between the two functions, as described.

Thus, the correct answer is D. (-1, ∞).

Would you like further details or clarifications?

Here are 5 related questions to expand on this:

1. How do we determine the number of solutions for a quadratic-exponential equation?
2. What is the significance of the base in the exponential expression $\left( \frac{1}{3} \right)^{|x|}$?
3. Can this method be generalized to other exponential or quadratic equations?
4. How does the value of $a$ shift the parabola vertically in the equation?
5. How would the solutions change if the base of the exponential function was greater than 1?

Tip: When dealing with equations involving absolute values, consider the behavior of both positive and negative cases of the variable to ensure full coverage.