Math Problem Statement

Given the function

​f(x)equals=5 x minus 95x−9.

​(a) Find

f Superscript negative 1f−1.

​(b) Graph f and

f Superscript negative 1f−1

in the same rectangular coordinate system.

​(c) Use interval notation to give the domain and the range of f and

f Superscript negative 1f−1.

Question content area bottom

Part 1

​(a)**** The inverse function is

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)equals=enter your response here.

​(Use integers or fractions for any numbers in the​ expression.)

Part 2

​(b) Choose the correct graph which shows​ f(x) and

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)

graphed in the same coordinate system. The graph of​ f(x) is shown as a solid line and the graph of

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)

is shown as a dotted line.

A.

-1010-1010xy

A coordinate plane has a horizontal x-axis and a vertical y-axis each labeled from negative 10 to 10 in increments of 1. There are two lines. The solid line increases rapidly from left to right, intersecting the y-axis at negative 9 and the x-axis approximately at 1.8. The dotted line increases slowly from left to right, intersecting the x-axis at negative 9 and the y-axis approximately at 1.8.

B.

-1010-1010xy

A coordinate plane has a horizontal x-axis and a vertical y-axis each labeled from negative 10 to 10 in increments of 1. There are two lines. The solid line increases rapidly from left to right, intersecting the y-axis at negative 9 and the x-axis approximately at 1.8. The dotted line decreases slowly from left to right, intersecting the x-axis at negative 9 and the y-axis approximately at negative 1.8.

C.

-1010-1010xy

A coordinate plane has a horizontal x-axis and a vertical y-axis each labeled from negative 10 to 10 in increments of 1. There are two lines. The solid line decreases slowly from left to right, intersecting the y-axis approximately at 1.8 and the x-axis at 9. The dotted line increases rapidly from left to right, intersecting the y-axis at negative 9 and the x-axis approximately at 1.8.

D.

-1010-1010xy

A coordinate plane has a horizontal x-axis and a vertical y-axis each labeled from negative 10 to 10 in increments of 1. There are two lines. The solid line decreases slowly from left to right, intersecting the y-axis approximately at 1.8 and the x-axis at 9. The dotted line decreases rapidly from left to right, intersecting the y-axis at 9 and the x-axis approximately at 1.8.

Part 3

​(c) State the domain and range of​ f(x) using interval notation.

The domain of​ f(x) is

enter your response here​,

and the range of​ f(x) is

enter your response here.

Part 4

State the domain and range of

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)

using interval notation.

The domain of

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)

is

enter your response here​,

and the range of

f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x)

is

enter your response here.

Solution

Part 1: Finding the inverse function f−1(x)f^{-1}(x)

Given the function:

f(x)=5x−9f(x) = 5x - 9

To find the inverse f−1(x)f^{-1}(x), follow these steps:

  1. Replace f(x)f(x) with yy: y=5x−9y = 5x - 9

  2. Swap xx and yy (to find the inverse): x=5y−9x = 5y - 9

  3. Solve for yy: x+9=5yx + 9 = 5y y=x+95y = \frac{x + 9}{5}

Thus, the inverse function is:

f−1(x)=x+95f^{-1}(x) = \frac{x + 9}{5}

Part 2: Graphing f(x)f(x) and f−1(x)f^{-1}(x)

We have two functions:

  • f(x)=5x−9f(x) = 5x - 9
  • f−1(x)=x+95f^{-1}(x) = \frac{x + 9}{5}

These graphs are reflections of each other across the line y=xy = x.

From the graph descriptions, the correct graph is Option A, where the solid line (representing f(x)f(x)) and the dotted line (representing f−1(x)f^{-1}(x)) intersect the axes appropriately.

  • f(x)f(x) is a solid line that increases rapidly, crossing the y-axis at (0,−9)(0, -9) and the x-axis around (1.8,0)(1.8, 0).
  • f−1(x)f^{-1}(x) is a dotted line that increases slowly, crossing the x-axis at (−9,0)(-9, 0) and the y-axis around (0,1.8)(0, 1.8).

Part 3: Domain and Range of f(x)f(x)

Since f(x)=5x−9f(x) = 5x - 9 is a linear function, its domain and range are both all real numbers.

  • The domain of f(x)f(x) is (−∞,∞)(-\infty, \infty).
  • The range of f(x)f(x) is (−∞,∞)(-\infty, \infty).

Part 4: Domain and Range of f−1(x)f^{-1}(x)

For the inverse function f−1(x)=x+95f^{-1}(x) = \frac{x + 9}{5}, its domain and range are also all real numbers since it is a linear function as well.

  • The domain of f−1(x)f^{-1}(x) is (−∞,∞)(-\infty, \infty).
  • The range of f−1(x)f^{-1}(x) is (−∞,∞)(-\infty, \infty).

Final Answer:

  1. (a) f−1(x)=x+95f^{-1}(x) = \frac{x + 9}{5}
  2. (b) The correct graph is Option A.
  3. (c) Domain and range:
    • Domain of f(x)f(x): (−∞,∞)(-\infty, \infty)
    • Range of f(x)f(x): (−∞,∞)(-\infty, \infty)
    • Domain of f−1(x)f^{-1}(x): (−∞,∞)(-\infty, \infty)
    • Range of f−1(x)f^{-1}(x): (−∞,∞)(-\infty, \infty)

Would you like more details on any part of the process? Here are some related questions to explore:

  1. How do you find the inverse of a more complicated function?
  2. What is the geometric relationship between a function and its inverse?
  3. How do you confirm that two functions are inverses of each other?
  4. What is the significance of the line y=xy = x in the context of inverses?
  5. Can every function have an inverse? Why or why not?

Tip: When finding the inverse of a function, always swap the variables and solve for the original dependent variable.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Inverse Functions
Graphing
Domain and Range

Formulas

Inverse function formula: swap x and y, then solve for y
f(x) = 5x - 9
f^{-1}(x) = (x + 9) / 5

Theorems

Inverse function theorem
Reflection property of inverse functions (about y = x)

Suitable Grade Level

Grades 9-11

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