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Algebra / Linear functions Difficulty: Easy

For the linear function $f$ , $f left parenthesis 0 right parenthesis equals 17$ and $f left parenthesis 1 right parenthesis equals 17$ . Which equation defines $f$ ?

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Explanation

Choice C is correct. An equation defining the linear function $f$ can be written in the form $f (x) = m x + b$ , where $m$ is the slope and $left parenthesis 0 comma b right parenthesis$ is the y-intercept of the graph of $y = f (x)$ in the xy-plane. The slope of the graph of $y = f (x)$ can be found using the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , where $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ are any two points that the graph passes through. If $f left parenthesis 0 right parenthesis equals 17$ , it follows that the graph of $y = f (x)$ passes through the point $left parenthesis 0 comma 17 right parenthesis$ . If $f left parenthesis 1 right parenthesis equals 17$ , it follows that the graph of $y = f (x)$ passes through the point $left parenthesis 1 comma 17 right parenthesis$ . Substituting $left parenthesis 0 comma 17 right parenthesis$ and $left parenthesis 1 comma 17 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ yields $m = \frac{17 - 17}{1 - 0}$ , which is equivalent to $m = \frac{0}{1}$ , or $m equals 0$ . Since the graph of $y = f (x)$ passes through $left parenthesis 0 comma 17 right parenthesis$ , it follows that $b equals 17$ . Substituting $0$ for $m$ and $17$ for $b$ in the equation $f (x) = m x + b$ yields $f (x) = 0 x + 17$ , or $f left parenthesis x right parenthesis equals 17$ . Thus, the equation that defines $f$ is $f left parenthesis x right parenthesis equals 17$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.