Math

Question Redefine a linear function y=6x+13y=6x+13, find its inverse, and determine the relationship between the functions.
f(x)=6x+13f(x) = 6x + 13 x=(y13)/6x = (y-13)/6 g(y)=(y13)/6g(y) = (y-13)/6 gg undoes the process of ff, f=g1f=g^{-1}, ff and gg are inverses.

Studdy Solution

STEP 1

Assumptions
1. The given formula is y=6x+13y=6x+13.
2. We are to define the rule for a function ff that determines yy in terms of xx.
3. We are to solve the equation y=6x+13y=6x+13 for xx.
4. We are to define the rule for a function gg that determines xx in terms of yy.
5. We need to determine the relationship between functions ff and gg.

STEP 2

Define the rule for the function ff that determines yy in terms of xx using the given formula.
f(x)=6x+13f(x) = 6x + 13

STEP 3

Solve the equation y=6x+13y=6x+13 for xx by isolating xx on one side of the equation.
y=6x+13y = 6x + 13

STEP 4

Subtract 13 from both sides of the equation to isolate the term containing xx.
y13=6xy - 13 = 6x

STEP 5

Divide both sides of the equation by 6 to solve for xx.
y136=x\frac{y - 13}{6} = x

STEP 6

Write the expression for xx in terms of yy.
x=y136x = \frac{y - 13}{6}

STEP 7

Define the rule for the function gg that determines xx in terms of yy using the expression found in STEP_6.
g(y)=y136g(y) = \frac{y - 13}{6}

STEP 8

Determine if gg undoes the process of ff by checking if applying gg after ff returns the original input xx.
g(f(x))=g(6x+13)g(f(x)) = g(6x + 13)

STEP 9

Substitute the expression for g(y)g(y) into the equation from STEP_8.
g(f(x))=(6x+13)136g(f(x)) = \frac{(6x + 13) - 13}{6}

STEP 10

Simplify the expression inside the parentheses.
g(f(x))=6x6g(f(x)) = \frac{6x}{6}

STEP 11

Simplify the fraction to get the original input xx.
g(f(x))=xg(f(x)) = x

STEP 12

Determine if f=g1f=g^{-1} by checking if ff undoes the process of gg.
f(g(y))=f(y136)f(g(y)) = f\left(\frac{y - 13}{6}\right)

STEP 13

Substitute the expression for f(x)f(x) into the equation from STEP_12.
f(g(y))=6(y136)+13f(g(y)) = 6\left(\frac{y - 13}{6}\right) + 13

STEP 14

Simplify the expression inside the parentheses and the multiplication by 6.
f(g(y))=(y13)+13f(g(y)) = (y - 13) + 13

STEP 15

Simplify the equation to get the original input yy.
f(g(y))=yf(g(y)) = y

STEP 16

Based on the results from STEP_11 and STEP_15, we can conclude that gg undoes the process of ff, and ff undoes the process of gg.

STEP 17

Since g(f(x))=xg(f(x)) = x and f(g(y))=yf(g(y)) = y, we can conclude that ff and gg are inverses of each other.

STEP 18

The statements that are true are: - gg undoes the process of ff. - ff and gg are inverses. - g=f1g=f^{-1}
The statement f=g1f=g^{-1} is not true because ff is the original function and gg is its inverse, not the other way around.

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