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A function $f(x)$ increases by $6$ over every unit interval in $x$ and $f(0) = 0$ . $\newline$ Which could be a function rule for $f(x)$ ? $\newline$ Choices: $\newline$ (A) $f(x) = x - 6$ $\newline$ (B) $f(x) = x + 6$ $\newline$ (C) $f(x) = 6x$ $\newline$ (D) $f(x) = -6x$

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Intermediate Value Theorem

Full solution

Q. A function

f(x)

increases by

6

over every unit interval in

x

and

f(0) = 0

.

\newline

Which could be a function rule for

f(x)

?

\newline

Choices:

\newline

(A)

f(x) = x - 6

\newline

(B)

f(x) = x + 6

\newline

(C)

f(x) = 6x

\newline

(D)

f(x) = -6x

Eliminate incorrect choices: Since $f(0) = 0$ , we can immediately eliminate choices (A) and (B) because $f(0)$ would not be $0$ for these functions.
Check choice (C): Now, let's check choice (C) $f(x) = 6x$ . If $f(x)$ increases by $6$ over every unit interval, then $f(1)$ should be $6$ . Substituting $x = 1$ into $f(x) = 6x$ gives $f(1) = 6(1) = 6$ .
Eliminate choice (D): Choice (D) $f(x) = -6x$ can be eliminated because if $x$ increases, $f(x)$ would decrease, which contradicts the information that $f(x)$ increases by $6$ over every unit interval.
Identify correct function rule: Therefore, the correct function rule for $f(x)$ is (C) $f(x) = 6x$ , which satisfies both conditions: $f(0) = 0$ and the function increases by $6$ over every unit interval.

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