Consider the function represented by the equation $y-6 x-9=0$. Which answer shows the equation written in function notation with $x$ as the independent variable?
A. $f(x)=6 x+9$
B. $f(x)=\frac{1}{6} x+\frac{3}{2}$
C. $f(y)=6 y+9$
D. $f(y)=\frac{1}{6} y+\frac{3}{2}$
Answer (1)
Start with the given equation: y − 6 x − 9 = 0 .
Solve for y in terms of x : y = 6 x + 9 .
Express the equation in function notation: f ( x ) = 6 x + 9 .
The equation in function notation is f ( x ) = 6 x + 9 .
Explanation
Understanding the Problem We are given the equation y − 6 x − 9 = 0 and we want to express it in function notation with x as the independent variable. This means we want to write the equation in the form y = f ( x ) , where f ( x ) is an expression involving x .
Solving for y To express the given equation in function notation, we need to solve for y in terms of x . Starting with the equation y − 6 x − 9 = 0 , we add 6 x and 9 to both sides to isolate y :
y − 6 x − 9 + 6 x + 9 = 0 + 6 x + 9 y = 6 x + 9
Expressing in Function Notation Now that we have y expressed in terms of x , we can write the equation in function notation by replacing y with f ( x ) . Thus, we have: f ( x ) = 6 x + 9
Final Answer Therefore, the equation in function notation with x as the independent variable is f ( x ) = 6 x + 9 .
Examples
In physics, the distance traveled by an object moving with constant acceleration can be modeled using a quadratic function. If the acceleration is constant, the relationship between time and distance can be expressed as d = f ( t ) = a t 2 + b t + c , where d is the distance, t is the time, and a , b , and c are constants. This function notation allows physicists to easily calculate the distance traveled at any given time by simply plugging the time value into the function.
