The question contains a function and two parts to solve:

Given the function:

$f(x) = 8 - 2 \cdot \log(0.5x + 6)$

Part a: Bepaal het domein, bereik en de asymptoot van deze functie.

This part asks to:

1. Determine the domain of the function.
2. Find the range of the function.
3. Determine the asymptote of the function.

Part b: Bereken exact het snijpunt van de grafiek van f met de x-as.

This part asks to find the exact point where the graph of the function intersects the x-axis, which is where $f(x) = 0$.

Let’s start with part a:

Part a

1. Domain

For the function $f(x) = 8 - 2 \cdot \log(0.5x + 6)$, we need the argument of the logarithm to be positive, as the logarithmic function is only defined for positive values.

$0.5x + 6 > 0$

Solving for $x$:

$0.5x > -6$

$x > -12$

So the domain of the function is $x > -12$.

2. Asymptote

The function involves a logarithmic term. The vertical asymptote occurs where the argument inside the logarithm becomes zero:

$0.5x + 6 = 0$

Solving for $x$:

$0.5x = -6$

$x = -12$

Thus, the vertical asymptote is at $x = -12$.

3. Range

The function $f(x) = 8 - 2 \cdot \log(0.5x + 6)$ is a transformation of the basic logarithmic function $\log(x)$. As the logarithmic function can take any real value, the range of the transformed function is all real numbers, $(-\infty, \infty)$.

Part b

For the x-intercept, we set $f(x) = 0$:

$8 - 2 \cdot \log(0.5x + 6) = 0$

Now, solving for $x$:

$2 \cdot \log(0.5x + 6) = 8$

$\log(0.5x + 6) = 4$

Now, converting the logarithmic equation to its exponential form:

$0.5x + 6 = 10^4$

$0.5x + 6 = 10000$

$0.5x = 9994$

$x = \frac{9994}{0.5} = 19988$

So, the x-intercept is $x = 19988$.

Would you like further details on any of these steps or have any questions?

Related questions:

1. How can you interpret the effect of the transformation $8 - 2 \cdot \log(x)$ on the basic logarithmic function?
2. How do you find the asymptote of other logarithmic functions?
3. Can you solve other logarithmic equations involving transformations?
4. What does the domain of a function tell you about its possible values?
5. How can you verify the solution for the x-intercept?

Tip: When dealing with logarithmic functions, always remember that the argument inside the logarithm must be positive for the function to be valid.