Math

QuestionFind the linear function ff such that f(35)=12f\left(\frac{3}{5}\right)=\frac{1}{2} and f(2)=4f(2)=4.

Studdy Solution

STEP 1

Assumptions1. The function ff is linear, which means it can be written in the form f(x)=mx+bf(x) = mx + b, where mm is the slope and bb is the y-intercept. . We are given two points on the line (35,1)(\frac{3}{5}, \frac{1}{}) and (,4)(,4).

STEP 2

First, we need to find the slope of the line. The slope is given by the formulam=y2y1x2x1m = \frac{y2 - y1}{x2 - x1}

STEP 3

Now, plug in the given values for x1x1, y1y1, x2x2, and y2y2 to calculate the slope.
m=12235m = \frac{ - \frac{1}{2}}{2 - \frac{3}{5}}

STEP 4

implify the numerator and denominator.
m=8212103m = \frac{\frac{8}{2} - \frac{1}{2}}{\frac{10}{} - \frac{3}{}}

STEP 5

Calculate the slope.
m=7275m = \frac{\frac{7}{2}}{\frac{7}{5}}

STEP 6

implify the slope by multiplying the numerator and denominator by5.
m=×52×5=3510=3.5m = \frac{ \times5}{2 \times5} = \frac{35}{10} =3.5

STEP 7

Now that we have the slope, we can find the y-intercept bb by rearranging the equation of the line to solve for bbb=ymxb = y - mx

STEP 8

Substitute one of the given points and the calculated slope into the equation to find bb. We'll use the point (35,12)(\frac{3}{5}, \frac{1}{2})b=123.5×35b = \frac{1}{2} -3.5 \times \frac{3}{5}

STEP 9

implify the equation.
b=221b = \frac{}{2} - \frac{21}{}

STEP 10

Calculate the y-intercept bb.
b=1610=.6b = -\frac{16}{10} = -.6

STEP 11

Now that we have the slope mm and the y-intercept bb, we can write the equation of the linef(x)=3.5x.6f(x) =3.5x -.6This is the linear function ff that has the given function values.

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