To analyze the function $f(x) = \left( \frac{1}{4} \right)^{-x + 2}$ and determine its domain and range:

1. Domain: The domain of an exponential function like this is all real numbers because there are no restrictions on the values $x$ can take. So, the domain is: $(-\infty, \infty)$

2. Range: For the function $f(x) = \left( \frac{1}{4} \right)^{-x + 2}$, observe that it’s an exponential function with a positive base raised to a variable power. As $x$ approaches positive infinity, the value of $f(x)$ becomes very large, and as $x$ approaches negative infinity, the value of $f(x)$ approaches zero. Therefore, the range of $f(x)$ is all positive real numbers: $(0, \infty)$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$

Would you like a further breakdown of how exponential functions affect domain and range, or do you have any questions?

Here are some related questions to explore this topic further:

1. How does the base of an exponential function affect its growth or decay?
2. What happens to the graph of an exponential function if we change the sign of the exponent?
3. How can we determine the horizontal asymptote of an exponential function?
4. What is the effect of shifting an exponential function up or down?
5. How do transformations, such as reflections, affect the range of an exponential function?

Tip: In exponential functions, when the base is a fraction (like $\frac{1}{4}$), the function decreases as $x$ increases.