Question

Part 1. Using the two functions listed below, insert numbers in place of the letters a, b, c, and d so that f(x) and g(x) are inverses.

f(x)=
x+a
b

g(x)=cx−d

Part 2. Show your work to prove that the inverse of f(x) is g(x).

Part 3. Show your work to evaluate g(f(x)).

Part 4. Graph your two functions on a coordinate plane. Include a table of values for each function. Include 5 values for each function. Graph the line y = x on the same graph.

Answer 1

@jdoe0001

Answer 2

@SedateFrog712

Answer 3

@aum

Answer 4

f(x) = x + a ??? 'b' is all by itself on a separate line. Where does it belong?

Answer 5

sorry thought i fixed it already
\[f(x)=\frac{ x+a }{ b }\]

Answer 6

Find the inverse of \[f(x)=\frac{ x+a }{ b } \\
y = \frac{ x+a }{ b } \\
\text{Switch x and y:} \\
x = \frac{ y+a }{ b } \\
\text{Solve for y:} \\
x * b = y + a \\
y = bx - a \\
f^{-1}(x) = bx - a \\
\text{Compare it to g(x): } \\
g(x) = cx - d \\
b = c \\
a = d
\]

Answer 7

followed you til we got to g(x)

Answer 8

The problem states g(x) is the inverse function and that g(x) = cx - d

Answer 9

In order for g(x) to be the inverse of \(f^{-1}(x)\), b must equal c and a must equal d.

Pick any values for c and d.
Example c = 2; d = 3. Then a = 3 and b = 2

Therefore, \(\large f(x)= \Large \frac{ x+3 }{ 2 } \) and \(\large g(x) = 2x - 3\) are inverses.

Answer 10

Part 2) To prove g(x) is the inverse of f(x), find f(g(x)) and prove it is x. If f(g(x)) = x then g(x) is the inverse of f(x).

Answer 11

Part 3) Evaluate g(f(x))

Part 4) Both f(x) and g(x) are straight lines. Just two points on each line will be sufficient to graph each line. But they want a table of five x values for each function. You can choose x = -2, -1, 0, 1, 2 and calculate f(x) and g(x) and put them in a table.
The line y = x will be a 45 degree straight line passing through the origin.
Since f(x) and g(x) are inverse of each other, their graphs should be symmetric with respect to the y = x line.