To define $f(x)$ as a linear function on the interval $[0, 2]$ so that it is continuous for all $x$, we need to ensure that:

1. $f(x)$ is continuous at $x = 0$ and $x = 2$.
2. The linear function $f(x)$ on $[0, 2]$ must fit into the overall function such that the limits from both sides match at these endpoints.

Step 1: Continuity at $x = 0$

The function $f(x) = 2x^2 - 1$ is already defined for $x < 0$. We can calculate $f(0)$ using this part of the function:

$f(0) = 2(0)^2 - 1 = -1$

Thus, to ensure continuity at $x = 0$, the linear function on $[0, 2]$ must pass through the point $(0, -1)$.

Step 2: Continuity at $x = 2$

The function $f(x) = 5x^2 - 3$ is defined for $x > 2$. We calculate the value at $x = 2$:

$f(2) = 5(2)^2 - 3 = 5(4) - 3 = 20 - 3 = 17$

Thus, the linear function on $[0, 2]$ must pass through the point $(2, 17)$.

Step 3: Defining the Linear Function

We now need to find a linear function $f(x) = mx + b$ that passes through the points $(0, -1)$ and $(2, 17)$.

• At $x = 0$, $f(0) = -1$, so $b = -1$.
• At $x = 2$, $f(2) = 17$, so we use the equation $f(x) = mx - 1$ and substitute $x = 2$:

$m(2) - 1 = 17$ $2m - 1 = 17$ $2m = 18$ $m = 9$

Thus, the linear function on $[0, 2]$ is:

$f(x) = 9x - 1$

Step 4: Verifying Continuity

Now, we verify the continuity of $f(x)$ over the entire domain:

• For $x < 0$, $f(x) = 2x^2 - 1$.
• For $0 \leq x \leq 2$, $f(x) = 9x - 1$.
• For $x > 2$, $f(x) = 5x^2 - 3$.

At $x = 0$, both $f(x) = 2x^2 - 1$ and $f(x) = 9x - 1$ give $f(0) = -1$, ensuring continuity.

At $x = 2$, both $f(x) = 9x - 1$ and $f(x) = 5x^2 - 3$ give $f(2) = 17$, ensuring continuity.

Thus, the function is continuous everywhere, and the linear function on $[0, 2]$ is $f(x) = 9x - 1$.

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Would you like further details or have any questions? Here are some related questions:

1. How do you handle piecewise continuous functions in calculus?
2. What are the conditions for a function to be continuous at a point?
3. How does the linear approximation work in piecewise functions?
4. Can you define a similar function where the linear section intersects the quadratic functions at different points?
5. What role do limits play in ensuring continuity in piecewise functions?

Tip: To ensure continuity in piecewise functions, always check the limits from both sides at the boundary points.