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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) $f(x) = 6^x$ $\newline$ (B) $f(x) = 6x$ $\newline$ (C) $f(x) = \frac{1}{6^x}$ $\newline$ (D) $f(x) = \frac{x}{6}$

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Identify linear and exponential functions

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) = 6^x

\newline

(B)

f(x) = 6x

\newline

(C)

f(x) = \frac{1}{6^x}

\newline

(D)

f(x) = \frac{x}{6}

Identify Function Type: Determine the type of function based on the rate of increase. Since $f(x)$ increases by $6$ over every unit interval in $x$ , this suggests a constant rate of change, which is characteristic of a linear function.
General Form of Linear Function: Write the general form of a linear function. $\newline$ The general form of a linear function is $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
Find Slope: Use the given information to find the slope $m$ . The function increases by $6$ for each unit increase in $x$ , so the slope $m$ is $6$ .
Find Y-Intercept: Use the given information to find the y-intercept $b$ . Since $f(0) = 0$ , when $x = 0$ , $f(x)$ should equal $0$ . Plugging these values into the general form gives us $0 = m(0) + b$ , which simplifies to $b = 0$ .
Write Specific Function Rule: Write the specific function rule for $f(x)$ . With $m = 6$ and $b = 0$ , the function rule is $f(x) = 6x$ .

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