Question

from which of the following functions is an avrage rate of change from x = -1 to x = 2 equal to 5

Answer 1

To find a function \( f(x) \) for which the average rate of change from \( x = -1 \) to \( x = 2 \) is equal to 5, we can use the formula for the average rate of change of a function:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Here, \( a = -1 \) and \( b = 2 \). So we have:

\[
\text{Average Rate of Change} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{f(2) - f(-1)}{3}
\]

We want this to equal 5:

\[
\frac{f(2) - f(-1)}{3} = 5
\]

Multiplying both sides by 3 gives:

\[
f(2) - f(-1) = 15
\]

Now, we need to find a function \( f(x) \) such that the difference \( f(2) - f(-1) = 15 \).

### Example Function

Let's consider a linear function of the form:

\[
f(x) = mx + b
\]

Compute \( f(2) \) and \( f(-1) \):

\[
f(2) = 2m + b
\]
\[
f(-1) = -m + b
\]

Now, substituting these into the equation:

\[
f(2) - f(-1) = (2m + b) - (-m + b) = 2m + b + m - b = 3m
\]

Setting this equal to 15:

\[
3m = 15 \\
m = 5
\]

So, we can have a function like:

\[
f(x) = 5x + b
\]

The value of \( b \) can be any real number, so one possible function is:

\[
f(x) = 5x
\]

### Verification with \( f(x) = 5x \)

Let's verify:

\[
f(2) = 5(2) = 10
\]
\[
f(-1) = 5(-1) = -5
\]
\[
f(2) - f(-1) = 10 - (-5) = 10 + 5 = 15
\]

Thus, the average rate of change from \( x = -1 \) to \( x = 2 \) is indeed:

\[
\frac{15}{3} = 5
\]

### Conclusion

One function that satisfies the condition is:

\[
f(x) = 5x
\]

You can also have other functions that take the form \( f(x) = 5x + b \) for any constant \( b \).