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Algebra / Linear functions Difficulty: Hard

The function $f (x)$ is defined as $19$ more than $4$ times a number $x$ . If $y = f (x)$ is graphed in the $x y$ -plane, what is the best interpretation of the $x$ -intercept?

When $f left parenthesis x right parenthesis equals 0$ , the number is $- \frac{19}{4}$ .

When the number is $0$ , $f left parenthesis x right parenthesis equals 19$ .

The value of $f (x)$ increases by $1$ for each increase of $4$ in the value of the number.

For each increase of $1$ in the value of the number, $f (x)$ increases by $4$ .

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Explanation

Choice A is correct. It’s given that the function $f (x)$ is defined as $19$ more than $4$ times a number $x$ . This can be represented by the equation $f (x) = 4 x + 19$ . The x-intercept of the graph of $y = f (x)$ in the xy-plane is the point where the graph intersects the x-axis, or the point on the graph where the value of $f (x)$ is equal to $0$ . Substituting $0$ for $f (x)$ in the equation $f (x) = 4 x + 19$ yields $0 = 4 x + 19$ . Subtracting $19$ from each side of this equation yields $- 19 = 4 x$ . Dividing each side of this equation by $4$ yields $x = - \frac{19}{4}$ . Therefore, when $f left parenthesis x right parenthesis equals 0$ , the number is $- \frac{19}{4}$ .

Choice B is incorrect. This is the best interpretation of the y-intercept, not the x-intercept, of the graph of the function.

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

Choice D is incorrect. This is the best interpretation of the slope, not the x-intercept, of the graph of the function.