To find the values of $y$ when $x = 1, 2, 3$ for an exponential function that passes through the points $(0, 5)$ and $(4, 25)$, we need to first determine the equation of the exponential function.

The general form of an exponential function is:

$y = ab^x$

where $a$ is the initial value (the value of $y$ when $x = 0$) and $b$ is the base (the growth or decay factor).

Step 1: Determine $a$

From the point $(0, 5)$, we know that:

$y = ab^0 = a = 5$

So, the equation becomes:

$y = 5b^x$

Step 2: Determine $b$

We now use the second point, $(4, 25)$, to find $b$. Substituting $x = 4$ and $y = 25$ into the equation:

$25 = 5b^4$

Dividing both sides by 5:

$b^4 = 5$

Taking the fourth root of both sides:

$b = \sqrt[4]{5}$

Step 3: Find $y$ for $x = 1, 2, 3$

Now that we have $a = 5$ and $b = \sqrt[4]{5}$, the equation is:

$y = 5 \left( \sqrt[4]{5} \right)^x$

We can now compute the values of $y$ for $x = 1, 2, 3$.

1. For $x = 1$:

[ y = 5 \times \left( \sqrt[4]{5} \right)^1 = 5 \times \sqrt[4]{5} ]

2. For $x = 2$:

$y = 5 \times \left( \sqrt[4]{5} \right)^2 = 5 \times \sqrt{5}$

3. For $x = 3$:

$y = 5 \times \left( \sqrt[4]{5} \right)^3 = 5 \times 5^{3/4}$

If you'd like, I can provide the exact numerical values for these calculations. Let me know!

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Would you like more detailed steps, or do you have any questions about this? Here are 5 questions to explore further:

1. How is the base $b$ of an exponential function related to the growth rate?
2. How would the equation change if the function were decaying rather than growing?
3. Can you use a logarithmic approach to solve for $b$ in an exponential function?
4. How would this method differ if the points represented a different type of curve, such as a linear function?
5. What happens to the function as $x$ increases indefinitely?

Tip: The initial value $a$ in an exponential function always represents the $y$-intercept (the value when $x = 0$).