Function Operations - Ninth Grade Math
Introduction to Function Operations
Function Operations: Mathematical operations performed on functions
Just like numbers: Functions can be added, subtracted, multiplied, and divided
Result: A new function is created
Notation: Operations use special notation to show combining functions
Just like numbers: Functions can be added, subtracted, multiplied, and divided
Result: A new function is created
Notation: Operations use special notation to show combining functions
Key Concept:
When we perform operations on functions, we combine their rules (formulas) to create a new function. The operations are performed on the function values, not just the function names.
When we perform operations on functions, we combine their rules (formulas) to create a new function. The operations are performed on the function values, not just the function names.
1. Add and Subtract Functions
Function Addition: Combining two functions by adding their outputs
Function Subtraction: Combining two functions by subtracting their outputs
Function Subtraction: Combining two functions by subtracting their outputs
Addition of Functions
Addition Formula:
$$(f + g)(x) = f(x) + g(x)$$
Read as: "f plus g of x equals f of x plus g of x"
Meaning: To find $(f + g)(x)$, add the expressions for $f(x)$ and $g(x)$
$$(f + g)(x) = f(x) + g(x)$$
Read as: "f plus g of x equals f of x plus g of x"
Meaning: To find $(f + g)(x)$, add the expressions for $f(x)$ and $g(x)$
Steps to Add Functions:
Step 1: Write out both function expressions
Step 2: Add the expressions together
Step 3: Combine like terms
Step 4: Simplify the result
Step 1: Write out both function expressions
Step 2: Add the expressions together
Step 3: Combine like terms
Step 4: Simplify the result
Example 1: Given $f(x) = 2x + 3$ and $g(x) = x - 5$, find $(f + g)(x)$
Solution:
$(f + g)(x) = f(x) + g(x)$
$= (2x + 3) + (x - 5)$
$= 2x + 3 + x - 5$
$= 3x - 2$
Answer: $(f + g)(x) = 3x - 2$
Solution:
$(f + g)(x) = f(x) + g(x)$
$= (2x + 3) + (x - 5)$
$= 2x + 3 + x - 5$
$= 3x - 2$
Answer: $(f + g)(x) = 3x - 2$
Example 2: Given $f(x) = x^2 + 4$ and $g(x) = 3x^2 - 2x + 1$, find $(f + g)(x)$
$(f + g)(x) = (x^2 + 4) + (3x^2 - 2x + 1)$
$= x^2 + 4 + 3x^2 - 2x + 1$
$= 4x^2 - 2x + 5$
Answer: $(f + g)(x) = 4x^2 - 2x + 5$
$(f + g)(x) = (x^2 + 4) + (3x^2 - 2x + 1)$
$= x^2 + 4 + 3x^2 - 2x + 1$
$= 4x^2 - 2x + 5$
Answer: $(f + g)(x) = 4x^2 - 2x + 5$
Example 3: If $f(x) = 5x$ and $g(x) = 2x + 7$, find $(f + g)(3)$
Method 1 - Find the combined function first:
$(f + g)(x) = 5x + (2x + 7) = 7x + 7$
$(f + g)(3) = 7(3) + 7 = 21 + 7 = 28$
Method 2 - Evaluate each function, then add:
$f(3) = 5(3) = 15$
$g(3) = 2(3) + 7 = 13$
$(f + g)(3) = 15 + 13 = 28$
Answer: 28
Method 1 - Find the combined function first:
$(f + g)(x) = 5x + (2x + 7) = 7x + 7$
$(f + g)(3) = 7(3) + 7 = 21 + 7 = 28$
Method 2 - Evaluate each function, then add:
$f(3) = 5(3) = 15$
$g(3) = 2(3) + 7 = 13$
$(f + g)(3) = 15 + 13 = 28$
Answer: 28
Subtraction of Functions
Subtraction Formula:
$$(f - g)(x) = f(x) - g(x)$$
Read as: "f minus g of x equals f of x minus g of x"
Important: Be careful with signs! Distribute the negative sign
$$(f - g)(x) = f(x) - g(x)$$
Read as: "f minus g of x equals f of x minus g of x"
Important: Be careful with signs! Distribute the negative sign
Steps to Subtract Functions:
Step 1: Write out both function expressions
Step 2: Subtract the second from the first (use parentheses!)
Step 3: Distribute the negative sign
Step 4: Combine like terms
Step 5: Simplify
Step 1: Write out both function expressions
Step 2: Subtract the second from the first (use parentheses!)
Step 3: Distribute the negative sign
Step 4: Combine like terms
Step 5: Simplify
Example 4: Given $f(x) = 5x + 8$ and $g(x) = 2x + 3$, find $(f - g)(x)$
Solution:
$(f - g)(x) = f(x) - g(x)$
$= (5x + 8) - (2x + 3)$
$= 5x + 8 - 2x - 3$
$= 3x + 5$
Answer: $(f - g)(x) = 3x + 5$
Solution:
$(f - g)(x) = f(x) - g(x)$
$= (5x + 8) - (2x + 3)$
$= 5x + 8 - 2x - 3$
$= 3x + 5$
Answer: $(f - g)(x) = 3x + 5$
Example 5: Given $f(x) = x^2 + 3x - 1$ and $g(x) = 2x^2 - x + 4$, find $(f - g)(x)$
$(f - g)(x) = (x^2 + 3x - 1) - (2x^2 - x + 4)$
$= x^2 + 3x - 1 - 2x^2 + x - 4$
$= -x^2 + 4x - 5$
Answer: $(f - g)(x) = -x^2 + 4x - 5$
$(f - g)(x) = (x^2 + 3x - 1) - (2x^2 - x + 4)$
$= x^2 + 3x - 1 - 2x^2 + x - 4$
$= -x^2 + 4x - 5$
Answer: $(f - g)(x) = -x^2 + 4x - 5$
Example 6: If $h(x) = 4x - 1$ and $k(x) = x + 5$, find $(h - k)(2)$
First find the combined function:
$(h - k)(x) = (4x - 1) - (x + 5)$
$= 4x - 1 - x - 5$
$= 3x - 6$
Now evaluate at $x = 2$:
$(h - k)(2) = 3(2) - 6 = 6 - 6 = 0$
Answer: 0
First find the combined function:
$(h - k)(x) = (4x - 1) - (x + 5)$
$= 4x - 1 - x - 5$
$= 3x - 6$
Now evaluate at $x = 2$:
$(h - k)(2) = 3(2) - 6 = 6 - 6 = 0$
Answer: 0
Common Mistakes to Avoid:
• Forgetting to distribute the negative sign in subtraction
• Not using parentheses around the second function
• Mixing up the order: $(f - g) \neq (g - f)$
• Forgetting to distribute the negative sign in subtraction
• Not using parentheses around the second function
• Mixing up the order: $(f - g) \neq (g - f)$
2. Multiply Functions
Function Multiplication: Combining two functions by multiplying their outputs
Result: Often a polynomial of higher degree
Result: Often a polynomial of higher degree
Multiplication Formula:
$$(f \cdot g)(x) = f(x) \cdot g(x)$$
Alternative Notation: $(fg)(x) = f(x) \cdot g(x)$
Read as: "f times g of x equals f of x times g of x"
Meaning: Multiply the expressions for $f(x)$ and $g(x)$
$$(f \cdot g)(x) = f(x) \cdot g(x)$$
Alternative Notation: $(fg)(x) = f(x) \cdot g(x)$
Read as: "f times g of x equals f of x times g of x"
Meaning: Multiply the expressions for $f(x)$ and $g(x)$
Steps to Multiply Functions:
Step 1: Write out both function expressions
Step 2: Multiply the expressions (use FOIL or distributive property)
Step 3: Expand completely
Step 4: Combine like terms
Step 5: Write in standard form
Step 1: Write out both function expressions
Step 2: Multiply the expressions (use FOIL or distributive property)
Step 3: Expand completely
Step 4: Combine like terms
Step 5: Write in standard form
Example 1: Given $f(x) = 3x$ and $g(x) = x + 2$, find $(f \cdot g)(x)$
Solution:
$(f \cdot g)(x) = f(x) \cdot g(x)$
$= (3x)(x + 2)$
$= 3x \cdot x + 3x \cdot 2$
$= 3x^2 + 6x$
Answer: $(f \cdot g)(x) = 3x^2 + 6x$
Solution:
$(f \cdot g)(x) = f(x) \cdot g(x)$
$= (3x)(x + 2)$
$= 3x \cdot x + 3x \cdot 2$
$= 3x^2 + 6x$
Answer: $(f \cdot g)(x) = 3x^2 + 6x$
Example 2: Given $f(x) = x + 3$ and $g(x) = x - 5$, find $(f \cdot g)(x)$
Use FOIL:
$(f \cdot g)(x) = (x + 3)(x - 5)$
$= x^2 - 5x + 3x - 15$
$= x^2 - 2x - 15$
Answer: $(f \cdot g)(x) = x^2 - 2x - 15$
Use FOIL:
$(f \cdot g)(x) = (x + 3)(x - 5)$
$= x^2 - 5x + 3x - 15$
$= x^2 - 2x - 15$
Answer: $(f \cdot g)(x) = x^2 - 2x - 15$
Example 3: Given $f(x) = 2x - 1$ and $g(x) = 3x + 4$, find $(f \cdot g)(x)$
$(f \cdot g)(x) = (2x - 1)(3x + 4)$
Using FOIL:
F: $2x \cdot 3x = 6x^2$
O: $2x \cdot 4 = 8x$
I: $-1 \cdot 3x = -3x$
L: $-1 \cdot 4 = -4$
$= 6x^2 + 8x - 3x - 4$
$= 6x^2 + 5x - 4$
Answer: $(f \cdot g)(x) = 6x^2 + 5x - 4$
$(f \cdot g)(x) = (2x - 1)(3x + 4)$
Using FOIL:
F: $2x \cdot 3x = 6x^2$
O: $2x \cdot 4 = 8x$
I: $-1 \cdot 3x = -3x$
L: $-1 \cdot 4 = -4$
$= 6x^2 + 8x - 3x - 4$
$= 6x^2 + 5x - 4$
Answer: $(f \cdot g)(x) = 6x^2 + 5x - 4$
Example 4: If $f(x) = x^2$ and $g(x) = 2x + 1$, find $(f \cdot g)(x)$
$(f \cdot g)(x) = x^2(2x + 1)$
$= 2x^3 + x^2$
Answer: $(f \cdot g)(x) = 2x^3 + x^2$
$(f \cdot g)(x) = x^2(2x + 1)$
$= 2x^3 + x^2$
Answer: $(f \cdot g)(x) = 2x^3 + x^2$
Example 5: Given $f(x) = 4x$ and $g(x) = x - 3$, find $(f \cdot g)(2)$
Method 1 - Find combined function first:
$(f \cdot g)(x) = 4x(x - 3) = 4x^2 - 12x$
$(f \cdot g)(2) = 4(2)^2 - 12(2) = 16 - 24 = -8$
Method 2 - Evaluate each, then multiply:
$f(2) = 4(2) = 8$
$g(2) = 2 - 3 = -1$
$(f \cdot g)(2) = 8 \cdot (-1) = -8$
Answer: -8
Method 1 - Find combined function first:
$(f \cdot g)(x) = 4x(x - 3) = 4x^2 - 12x$
$(f \cdot g)(2) = 4(2)^2 - 12(2) = 16 - 24 = -8$
Method 2 - Evaluate each, then multiply:
$f(2) = 4(2) = 8$
$g(2) = 2 - 3 = -1$
$(f \cdot g)(2) = 8 \cdot (-1) = -8$
Answer: -8
Key Points:
• Multiplication often increases the degree of the polynomial
• Use FOIL for binomials, distributive property for others
• Always simplify completely and combine like terms
• Multiplication often increases the degree of the polynomial
• Use FOIL for binomials, distributive property for others
• Always simplify completely and combine like terms
3. Find the Inverse of a Linear Function
Inverse Function: A function that "undoes" another function
Notation: $f^{-1}(x)$ (read as "f inverse of x")
Important: $f^{-1}$ does NOT mean $\frac{1}{f}$!
Key Property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$
Notation: $f^{-1}(x)$ (read as "f inverse of x")
Important: $f^{-1}$ does NOT mean $\frac{1}{f}$!
Key Property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$
What Does an Inverse Do?
If function $f$ takes input $a$ and gives output $b$:
$$f(a) = b$$
Then the inverse function $f^{-1}$ takes input $b$ and gives output $a$:
$$f^{-1}(b) = a$$
In other words: Input and output swap roles!
If $(x, y)$ is on $f$, then $(y, x)$ is on $f^{-1}$
If function $f$ takes input $a$ and gives output $b$:
$$f(a) = b$$
Then the inverse function $f^{-1}$ takes input $b$ and gives output $a$:
$$f^{-1}(b) = a$$
In other words: Input and output swap roles!
If $(x, y)$ is on $f$, then $(y, x)$ is on $f^{-1}$
Steps to Find Inverse of Linear Function:
Step 1: Replace $f(x)$ with $y$
Step 2: Swap $x$ and $y$ (interchange variables)
Step 3: Solve for $y$
Step 4: Replace $y$ with $f^{-1}(x)$
Step 5: Verify (optional but recommended)
Step 1: Replace $f(x)$ with $y$
Step 2: Swap $x$ and $y$ (interchange variables)
Step 3: Solve for $y$
Step 4: Replace $y$ with $f^{-1}(x)$
Step 5: Verify (optional but recommended)
Example 1: Find the inverse of $f(x) = 2x + 3$
Step 1: Replace $f(x)$ with $y$
$y = 2x + 3$
Step 2: Swap $x$ and $y$
$x = 2y + 3$
Step 3: Solve for $y$
$x - 3 = 2y$
$y = \frac{x - 3}{2}$
Step 4: Replace $y$ with $f^{-1}(x)$
$f^{-1}(x) = \frac{x - 3}{2}$
Answer: $f^{-1}(x) = \frac{x - 3}{2}$
Step 1: Replace $f(x)$ with $y$
$y = 2x + 3$
Step 2: Swap $x$ and $y$
$x = 2y + 3$
Step 3: Solve for $y$
$x - 3 = 2y$
$y = \frac{x - 3}{2}$
Step 4: Replace $y$ with $f^{-1}(x)$
$f^{-1}(x) = \frac{x - 3}{2}$
Answer: $f^{-1}(x) = \frac{x - 3}{2}$
Example 2: Find the inverse of $g(x) = \frac{x + 5}{3}$
Step 1: $y = \frac{x + 5}{3}$
Step 2: $x = \frac{y + 5}{3}$
Step 3: Solve for $y$
$3x = y + 5$
$y = 3x - 5$
Step 4: $g^{-1}(x) = 3x - 5$
Answer: $g^{-1}(x) = 3x - 5$
Step 1: $y = \frac{x + 5}{3}$
Step 2: $x = \frac{y + 5}{3}$
Step 3: Solve for $y$
$3x = y + 5$
$y = 3x - 5$
Step 4: $g^{-1}(x) = 3x - 5$
Answer: $g^{-1}(x) = 3x - 5$
Example 3: Find the inverse of $h(x) = -4x + 7$
$y = -4x + 7$
$x = -4y + 7$
$x - 7 = -4y$
$y = \frac{x - 7}{-4} = \frac{7 - x}{4}$
Answer: $h^{-1}(x) = \frac{7 - x}{4}$ or $\frac{-x + 7}{4}$
$y = -4x + 7$
$x = -4y + 7$
$x - 7 = -4y$
$y = \frac{x - 7}{-4} = \frac{7 - x}{4}$
Answer: $h^{-1}(x) = \frac{7 - x}{4}$ or $\frac{-x + 7}{4}$
Example 4: Find the inverse of $f(x) = x$ (identity function)
$y = x$
$x = y$ (swap)
$y = x$ (already solved!)
Answer: $f^{-1}(x) = x$
The identity function is its own inverse!
$y = x$
$x = y$ (swap)
$y = x$ (already solved!)
Answer: $f^{-1}(x) = x$
The identity function is its own inverse!
Verifying an Inverse
Verification Method:
To verify $f^{-1}$ is the inverse of $f$, check BOTH:
1. Composition Test:
$$f(f^{-1}(x)) = x$$
$$f^{-1}(f(x)) = x$$
Both must equal $x$ for all values!
To verify $f^{-1}$ is the inverse of $f$, check BOTH:
1. Composition Test:
$$f(f^{-1}(x)) = x$$
$$f^{-1}(f(x)) = x$$
Both must equal $x$ for all values!
Example 5: Verify that $f(x) = 3x - 6$ and $f^{-1}(x) = \frac{x + 6}{3}$ are inverses
Check $f(f^{-1}(x))$:
$f(f^{-1}(x)) = f\left(\frac{x + 6}{3}\right)$
$= 3\left(\frac{x + 6}{3}\right) - 6$
$= (x + 6) - 6$
$= x$ ✓
Check $f^{-1}(f(x))$:
$f^{-1}(f(x)) = f^{-1}(3x - 6)$
$= \frac{(3x - 6) + 6}{3}$
$= \frac{3x}{3}$
$= x$ ✓
Both equal $x$, so they are inverses!
Check $f(f^{-1}(x))$:
$f(f^{-1}(x)) = f\left(\frac{x + 6}{3}\right)$
$= 3\left(\frac{x + 6}{3}\right) - 6$
$= (x + 6) - 6$
$= x$ ✓
Check $f^{-1}(f(x))$:
$f^{-1}(f(x)) = f^{-1}(3x - 6)$
$= \frac{(3x - 6) + 6}{3}$
$= \frac{3x}{3}$
$= x$ ✓
Both equal $x$, so they are inverses!
Graphical Interpretation
Graph Properties of Inverse Functions:
• Graphs of $f$ and $f^{-1}$ are symmetric about the line $y = x$
• If you fold the graph along $y = x$, $f$ and $f^{-1}$ overlap
• Points $(a, b)$ on $f$ correspond to points $(b, a)$ on $f^{-1}$
• Graphs of $f$ and $f^{-1}$ are symmetric about the line $y = x$
• If you fold the graph along $y = x$, $f$ and $f^{-1}$ overlap
• Points $(a, b)$ on $f$ correspond to points $(b, a)$ on $f^{-1}$
Example 6: If $f(2) = 7$, what is $f^{-1}(7)$?
Since $f(2) = 7$:
Point $(2, 7)$ is on the graph of $f$
For inverse:
Point $(7, 2)$ is on the graph of $f^{-1}$
Therefore: $f^{-1}(7) = 2$
Since $f(2) = 7$:
Point $(2, 7)$ is on the graph of $f$
For inverse:
Point $(7, 2)$ is on the graph of $f^{-1}$
Therefore: $f^{-1}(7) = 2$
Important Notes:
• Only one-to-one functions have inverses
• All linear functions (except horizontal lines) have inverses
• The inverse "undoes" the original function
• Domain of $f$ = Range of $f^{-1}$
• Range of $f$ = Domain of $f^{-1}$
• Only one-to-one functions have inverses
• All linear functions (except horizontal lines) have inverses
• The inverse "undoes" the original function
• Domain of $f$ = Range of $f^{-1}$
• Range of $f$ = Domain of $f^{-1}$
Function Operations Summary
| Operation | Notation | Formula | Example |
|---|---|---|---|
| Addition | $(f + g)(x)$ | $f(x) + g(x)$ | If $f(x) = 2x$ and $g(x) = 3$ $(f + g)(x) = 2x + 3$ |
| Subtraction | $(f - g)(x)$ | $f(x) - g(x)$ | If $f(x) = 5x$ and $g(x) = x + 1$ $(f - g)(x) = 4x - 1$ |
| Multiplication | $(f \cdot g)(x)$ | $f(x) \cdot g(x)$ | If $f(x) = x$ and $g(x) = x + 2$ $(f \cdot g)(x) = x^2 + 2x$ |
| Division | $\left(\frac{f}{g}\right)(x)$ | $\frac{f(x)}{g(x)}$, $g(x) \neq 0$ | If $f(x) = 6x$ and $g(x) = 2$ $\left(\frac{f}{g}\right)(x) = 3x$ |
Steps for Finding Inverse Functions
| Step | Action | Example: $f(x) = 4x - 8$ |
|---|---|---|
| 1 | Replace $f(x)$ with $y$ | $y = 4x - 8$ |
| 2 | Swap $x$ and $y$ | $x = 4y - 8$ |
| 3 | Solve for $y$ | $x + 8 = 4y$ $y = \frac{x + 8}{4}$ |
| 4 | Replace $y$ with $f^{-1}(x)$ | $f^{-1}(x) = \frac{x + 8}{4}$ |
| 5 | Verify (optional) | Check: $f(f^{-1}(x)) = x$ |
Quick Reference Guide
When to Use Each Operation:
Addition $(f + g)$:
• Combining quantities
• Total cost, total distance, combined rates
Subtraction $(f - g)$:
• Finding differences
• Profit (revenue - cost), change in value
Multiplication $(f \cdot g)$:
• Area problems
• Scaling quantities
• Rate × Time = Distance type problems
Inverse $f^{-1}$:
• Reversing a process
• Converting units
• Finding input from output
Addition $(f + g)$:
• Combining quantities
• Total cost, total distance, combined rates
Subtraction $(f - g)$:
• Finding differences
• Profit (revenue - cost), change in value
Multiplication $(f \cdot g)$:
• Area problems
• Scaling quantities
• Rate × Time = Distance type problems
Inverse $f^{-1}$:
• Reversing a process
• Converting units
• Finding input from output
Common Applications
Application 1 - Business:
Revenue: $R(x) = 50x$ (50 per item)
Cost: $C(x) = 20x + 500$ (20 per item plus $500 fixed)
Profit Function:
$P(x) = R(x) - C(x)$
$P(x) = 50x - (20x + 500)$
$P(x) = 30x - 500$
Revenue: $R(x) = 50x$ (50 per item)
Cost: $C(x) = 20x + 500$ (20 per item plus $500 fixed)
Profit Function:
$P(x) = R(x) - C(x)$
$P(x) = 50x - (20x + 500)$
$P(x) = 30x - 500$
Application 2 - Temperature Conversion:
Celsius to Fahrenheit: $F(C) = \frac{9}{5}C + 32$
Find Inverse (Fahrenheit to Celsius):
$F = \frac{9}{5}C + 32$
$C = \frac{9}{5}F + 32$ (swap)
$C - 32 = \frac{9}{5}F$
$F = \frac{5}{9}(C - 32)$
Inverse: $C(F) = \frac{5}{9}(F - 32)$
Celsius to Fahrenheit: $F(C) = \frac{9}{5}C + 32$
Find Inverse (Fahrenheit to Celsius):
$F = \frac{9}{5}C + 32$
$C = \frac{9}{5}F + 32$ (swap)
$C - 32 = \frac{9}{5}F$
$F = \frac{5}{9}(C - 32)$
Inverse: $C(F) = \frac{5}{9}(F - 32)$
Success Tips for Function Operations:
✓ Always use parentheses when combining functions
✓ Distribute negative signs carefully in subtraction
✓ Use FOIL or distributive property for multiplication
✓ Combine like terms after operations
✓ For inverses: swap variables, then solve
✓ Verify inverses by checking $f(f^{-1}(x)) = x$
✓ Remember: $f^{-1}$ does NOT mean $\frac{1}{f}$
✓ Graphs of $f$ and $f^{-1}$ are symmetric about $y = x$
✓ Domain of $f$ becomes range of $f^{-1}$ and vice versa
✓ Practice with simple functions first!
✓ Always use parentheses when combining functions
✓ Distribute negative signs carefully in subtraction
✓ Use FOIL or distributive property for multiplication
✓ Combine like terms after operations
✓ For inverses: swap variables, then solve
✓ Verify inverses by checking $f(f^{-1}(x)) = x$
✓ Remember: $f^{-1}$ does NOT mean $\frac{1}{f}$
✓ Graphs of $f$ and $f^{-1}$ are symmetric about $y = x$
✓ Domain of $f$ becomes range of $f^{-1}$ and vice versa
✓ Practice with simple functions first!