https://file.aiquickdraw.com/mathbot/file/1726397471662j1lfo1rt.jpg

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.

If (1/3)^(|x|) = x^2 - a has two solutions, then the range of 'a' is equal to.

Explore how to determine the range of 'a' for the equation (1/3)^(|x|) = x^2 - a, ensuring two solutions. Understand the interplay between exponential and quadratic functions, absolute values, and solving for the range.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation