Let's break down the tasks based on the image you provided, which consists of two exercises: 4.9 and 4.10.

4.9

This question refers to a graph of a function $f$ with the domain $D_f = [-1, 5]$.

a. Finn nullpunktene til $f$

Find the zeros of the function $f$ (the points where the graph intersects the x-axis).

• From the graph, the function crosses the x-axis at $x = 0$ and $x = 4$.
• Therefore, the zeros of $f$ are $x = 0$ and $x = 4$.

b. Hva er verdimengden til $f$?

Find the range of the function $f$.

• From the graph, the minimum value of $f(x)$ is $-4$ (at $x \approx 3$) and the maximum value is $10$ (at $x = -1$).
• Thus, the range of $f$ is $[-4, 10]$.

c. Finn $f(0)$

Find the value of $f(0)$.

• Looking at the graph, when $x = 0$, $f(x) = 4$.
• Therefore, $f(0) = 4$.

d. For hvilke verdier av $x$ er $f(x) = 4$?

Find the values of $x$ for which $f(x) = 4$.

• From the graph, $f(x) = 4$ at $x = 0$ and $x = 2$.
• Therefore, the values of $x$ where $f(x) = 4$ are $x = 0$ and $x = 2$.

4.10

The functions $f(x) = x^2 - 4$ and $g(x) = 4 - x^2$ are given.

a. Finn eventuelle toppunkter og bunnpunkter på grafene

Find the possible maximum and minimum points on the graphs.

• For $f(x) = x^2 - 4$:

  • This is a parabola opening upwards. The vertex, which is the minimum point, is at $x = 0$, and $f(0) = -4$. Therefore, the minimum point is $(0, -4)$.

• For $g(x) = 4 - x^2$:

  • This is a parabola opening downwards. The vertex, which is the maximum point, is at $x = 0$, and $g(0) = 4$. Therefore, the maximum point is $(0, 4)$.

b. Finn skjæringspunktene mellom grafene

Find the intersection points of the graphs.

• To find where $f(x) = g(x)$, solve $x^2 - 4 = 4 - x^2$. $2x^2 = 8 \quad \Rightarrow \quad x^2 = 4 \quad \Rightarrow \quad x = \pm 2$
  • For $x = 2$, $f(2) = g(2) = 0$, and for $x = -2$, $f(-2) = g(-2) = 0$.
  • Therefore, the intersection points are $(2, 0)$ and $(-2, 0)$.

c. Finn nullpunktene til funksjonene

Find the zeros of the functions.

• For $f(x) = x^2 - 4$, solve $x^2 - 4 = 0$. The solutions are $x = \pm 2$.

  • The zeros of $f(x)$ are $x = 2$ and $x = -2$.

• For $g(x) = 4 - x^2$, solve $4 - x^2 = 0$. The solutions are also $x = \pm 2$.

  • The zeros of $g(x)$ are $x = 2$ and $x = -2$.

Do you need more details or clarifications on any of these points?
Here are some additional questions that might expand on the above content:

1. What are the exact coordinates of the minimum and maximum points for other quadratic functions?
2. How can you verify the intersection points of two functions algebraically?
3. Can you explore the vertex form of a quadratic equation to analyze the function behavior?
4. What is the significance of the range of a function in real-world applications?
5. How do the zeros of a function relate to its graphical representation?

Tip: Remember that the zeros of a function are the points where the graph intersects the x-axis. These points give valuable information about the behavior of the function.