Function evaluation: The process of finding the output value of a function for a specific input value
Note: In function notation f(x), we substitute the given value for x and simplify using order of operations.
- Identify the function rule (the expression after f(x) = )
- Replace every instance of x with the given input value
- Simplify the expression using order of operations
- Record the result as the function output
f(x) = 2x + 3
f(5) = 2(5) + 3
f(5) = 10 + 3 = 13
f(5) = 13
• Substitution: Replace x with the given input value
• Order of operations: Perform operations in correct sequence
• Simplification: Calculate the final result accurately
• Practice Tip: Always perform operations in the correct order: parentheses, exponents, multiplication/division, addition/subtraction
- If g(x) = x + 7, then g(3) = 3 + 7 = 10
- If h(x) = 4x - 1, then h(2) = 4(2) - 1 = 7
- If p(x) = -x + 5, then p(-2) = -(-2) + 5 = 7
- Think of f(x) as a machine that takes input x and produces output f(x)
- Always substitute the input value in place of every occurrence of x
- Be careful with negative inputs and negative coefficients
Q: What does f(x) mean?
A: f(x) means "the value of function f when x is the input" - it's another way of writing y.
Q: Can I have multiple functions like f(x), g(x), h(x)?
A: Yes, different letters represent different functions: f(x), g(x), h(x), etc.
Linear function: A function of the form f(x) = mx + b, where m and b are constants
Note: Linear functions have a constant rate of change and produce straight lines when graphed.
- Identify the function rule: f(x) = 3x - 7
- For each input value, substitute into the function rule
- Follow order of operations: multiplication, then addition/subtraction
- Calculate and record each output value
f(0) = 3(0) - 7 = 0 - 7 = -7
f(2) = 3(2) - 7 = 6 - 7 = -1
f(-1) = 3(-1) - 7 = -3 - 7 = -10
f(0) = -7, f(2) = -1, f(-1) = -10
• Substitution: Replace x with the given input value
• Order of operations: Perform operations in correct sequence
• Sign handling: Be careful with negative inputs
• Practice Tip: Work systematically through each value to avoid errors
- If g(x) = 2x + 5, then g(0) = 5, g(1) = 7, g(-1) = 3
- If h(x) = -x + 4, then h(0) = 4, h(2) = 2, h(-3) = 7
- If p(x) = 4x - 3, then p(0) = -3, p(1) = 1, p(-2) = -11
- When x = 0, the result is just the constant term (b in f(x) = mx + b)
- Work through each evaluation systematically to avoid confusion
- Check your arithmetic carefully, especially with negative values
Q: What happens when x = 0 in a linear function?
A: The result is just the constant term. For f(x) = mx + b, f(0) = b.
Q: How do I check my answers?
A: Substitute the input back into the original function and verify the calculation.
Quadratic function: A function of the form f(x) = ax² + bx + c, where a ≠ 0
Note: Quadratic functions have degree 2 and produce parabolic graphs when plotted.
- Identify the function rule: h(x) = x² + 2x - 3
- For each input value, substitute into the function rule
- Calculate each term separately: x², 2x, and -3
- Combine the terms to get the final result
h(3) = (3)² + 2(3) - 3 = 9 + 6 - 3 = 12
h(-3) = (-3)² + 2(-3) - 3 = 9 - 6 - 3 = 0
h(3) = 9 + 6 - 3 = 12 ✓
h(-3) = 9 - 6 - 3 = 0 ✓
h(3) = 12 and h(-3) = 0
• Quadratic evaluation: Calculate each term separately (x², bx, c)
• Sign handling: Pay attention to signs when substituting negative values
• Order of operations: Exponents first, then multiplication, then addition/subtraction
• Practice Tip: Remember that (-a)² = a² regardless of the sign of a
- If f(x) = x² - 4, then f(2) = 0 and f(-2) = 0
- If g(x) = x² + x + 1, then g(0) = 1 and g(-1) = 1
- If p(x) = 2x² - 3x + 1, then p(1) = 0 and p(-1) = 6
- Quadratic functions often have symmetry around their vertex
- Squaring negative numbers always gives positive results
- Pay attention to the signs of each term when substituting
Q: What if the quadratic has a coefficient for x²?
A: Multiply the coefficient by the squared term. For f(x) = 2x² + x - 1, f(3) = 2(9) + 3 - 1 = 20.
Q: Can quadratic functions have the same output for different inputs?
A: Yes, due to their symmetric nature, quadratics can have f(a) = f(b) for a ≠ b.
Function evaluation: Replacing the variable in a function with a specific value
Note: When substituting negative values, pay special attention to the signs of each term.
- Identify the function rule: g(x) = x² - 4x + 1
- Replace every x with the given input value (-2)
- Calculate each term separately, being careful with signs
- Combine the terms to get the final result
g(x) = x² - 4x + 1
g(-2) = (-2)² - 4(-2) + 1
(-2)² = 4, -4(-2) = 8, +1 = 1
g(-2) = 4 + 8 + 1 = 13
g(-2) = 13
• Sign handling: Be careful with negative inputs, especially when raising to powers
• Order of operations: Calculate exponents first, then multiplication, then addition/subtraction
• Systematic calculation: Work through each term separately
• Practice Tip: Remember (-a)² = a², but (-a)³ = -a³
- If h(x) = x² + 3x - 2, then h(-1) = 1 - 3 - 2 = -4
- If p(x) = -x² + 2x + 5, then p(3) = -9 + 6 + 5 = 2
- If q(x) = x² - 6x + 9, then q(3) = 9 - 18 + 9 = 0
- Pay special attention to signs when substituting negative values
- Calculate each term separately to avoid sign errors
- Remember that squaring a negative number gives a positive result
Q: What if the function has multiple terms with x?
A: Substitute the input value for every occurrence of x, then calculate each term separately.
Q: How do I handle negative inputs with exponents?
A: (-a)² = a² (positive), but (-a)³ = -a³ (negative). The result depends on whether the exponent is even or odd.
Function operations: Performing arithmetic operations on function values
Note: We can add, subtract, multiply, or divide function values just like regular numbers.
- Evaluate each function separately at the given input
- Perform the arithmetic operation on the results
- Combine to get the final answer
f(4) = 2(4) + 1 = 8 + 1 = 9
g(4) = 4 - 3 = 1
f(4) + g(4) = 9 + 1 = 10
f(4) + g(4) = 10
• Separate evaluation: Calculate each function value independently
• Arithmetic operations: Perform operations on the results
• Order of operations: Follow proper sequence when evaluating
• Practice Tip: Evaluate each function completely before performing operations
- If f(x) = x + 2 and g(x) = x - 1, then f(3) - g(3) = 5 - 2 = 3
- If p(x) = 2x and q(x) = x + 1, then p(2) × q(2) = 4 × 3 = 12
- If r(x) = x² and s(x) = x, then r(3) ÷ s(3) = 9 ÷ 3 = 3
- Evaluate each function separately before performing operations
- Keep track of which function you're evaluating at each step
- Double-check your arithmetic after evaluating each function
Q: Can I combine functions symbolically first?
A: Yes, you could define (f+g)(x) = f(x) + g(x) = (2x+1) + (x-3) = 3x-2, then evaluate (f+g)(4) = 10.
Q: What if I need to multiply function values?
A: Same approach: evaluate each function separately, then multiply the results.