The function $f$ is defined as $f(x)=2 x-3$ . $\newline$ What is the $x$ -coordinate of the point on the function's graph that is closest to the origin?

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Algebra 2

Rational functions: asymptotes and excluded values

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Q. The function

f

is defined as

f(x)=2 x-3

\newline

What is the

x

-coordinate of the point on the function's graph that is closest to the origin?

Define Distance Formula: To find the point closest to the origin, we need to minimize the distance from the point $(x, f(x))$ to the origin $(0,0)$ .
Substitute $f(x)$ : The distance $D$ from $(x, f(x))$ to the origin is given by the formula $D = \sqrt{x^2 + (f(x))^2}$ .
Expand Formula: Substitute $f(x)$ with $2x - 3$ into the distance formula: $D = \sqrt{x^2 + (2x - 3)^2}$ .
Combine Like Terms: Expand the formula: $D = \sqrt{x^2 + 4x^2 - 12x + 9}$ .
Take Derivative of D: Combine like terms: $D = \sqrt{5x^2 - 12x + 9}$ .
Minimize $D^2$ : To minimize the distance, we take the derivative of $D$ with respect to $x$ and set it to zero.
Find Critical Point: The derivative of $D$ is complicated because of the square root; instead, we can minimize $D^2$ to make it easier.
Solve for $x$ : Minimize $D^2 = 5x^2 - 12x + 9$ by taking the derivative and setting it to zero: $\frac{d(D^2)}{dx} = 10x - 12$ .
Simplify Fraction: Set the derivative equal to zero to find the critical point: $10x - 12 = 0$ .
Simplify Fraction: Set the derivative equal to zero to find the critical point: $10x - 12 = 0$ . Solve for $x$ : $10x = 12$ .
Simplify Fraction: Set the derivative equal to zero to find the critical point: $10x - 12 = 0$ . Solve for $x$ : $10x = 12$ . Divide by $10$ : $x = \frac{12}{10}$ .
Simplify Fraction: Set the derivative equal to zero to find the critical point: $10x - 12 = 0$ . Solve for $x$ : $10x = 12$ . Divide by $10$ : $x = \frac{12}{10}$ . Simplify the fraction: $x = 1.2$ .