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Unit 7: Exponential and General Functions

Topic 3

Function Notation and Evaluation

A function is a rule that assigns each input xx exactly one output yy. We often write functions using function notation:
y=f(x)y = f(x)
which means “ff of xx.” This notation emphasizes that yy depends on xx.
To evaluate a function, substitute a given value for xx and simplify.
Example:
f(x)=3x+2,f(4)=3(4)+2=14.f(x) = 3x + 2, \quad f(4) = 3(4) + 2 = 14.
The same rule applies for more complex expressions or multiple steps. SAT questions often ask for f(a)f(a), f(x+1)f(x+1), or f(2x)f(2x), which require substitution and simplification rather than solving.

Core Skills

  • Interpret f(x)f(x) as the value of the function at input xx.
  • Substitute specific or algebraic values into function definitions.
  • Simplify expressions like f(x+h)f(x+h) or f(2x)f(2x).
  • Understand f(x)f(x) vs. f(a)f(a) as general rule vs. specific value.
  • Recognize and evaluate composite expressions inside functions.

Example 1: Basic Evaluation

Given f(x)=2x+5,f(x) = 2x + 5, find f(3).f(3).
f(3)=2(3)+5=6+5=11.f(3) = 2(3) + 5 = 6 + 5 = 11.
f(3)=11\boxed{f(3) = 11}

Example 2: Substituting an Expression

If f(x)=4x−7,f(x) = 4x - 7, find f(x+2).f(x + 2).
f(x+2)=4(x+2)−7=4x+8−7=4x+1.f(x + 2) = 4(x + 2) - 7 = 4x + 8 - 7 = 4x + 1.
f(x+2)=4x+1\boxed{f(x + 2) = 4x + 1}

Example 3: Function in Context

A taxi fare is modeled by C(t)=3t+5,C(t) = 3t + 5, where tt is miles. Find C(10)C(10).
C(10)=3(10)+5=35.C(10) = 3(10) + 5 = 35.
The cost for 10 miles is $35.\boxed{\$35}.

Example 4: Evaluating a Quadratic Function

Given f(x)=x2−4x+1,f(x) = x^2 - 4x + 1, find f(−2)f(-2).
f(−2)=(−2)2−4(−2)+1=4+8+1=13.f(-2) = (-2)^2 - 4(-2) + 1 = 4 + 8 + 1 = 13.
f(−2)=13\boxed{f(-2) = 13}

Example 5: Algebraic Substitution

Given f(x)=3x2+2x−1,f(x) = 3x^2 + 2x - 1, find f(2x).f(2x).
f(2x)=3(2x)2+2(2x)−1=3(4x2)+4x−1=12x2+4x−1.f(2x) = 3(2x)^2 + 2(2x) - 1 = 3(4x^2) + 4x - 1 = 12x^2 + 4x - 1.
f(2x)=12x2+4x−1\boxed{f(2x) = 12x^2 + 4x - 1}

Example 6: Working with Multiple Functions

If f(x)=x+2f(x) = x + 2 and g(x)=3x−1,g(x) = 3x - 1, find:
f(3),g(3),f(g(3)).f(3), \quad g(3), \quad f(g(3)).
Compute step by step:
g(3)=3(3)−1=8.g(3) = 3(3) - 1 = 8.
f(3)=3+2=5.f(3) = 3 + 2 = 5.
f(g(3))=f(8)=8+2=10.f(g(3)) = f(8) = 8 + 2 = 10.
f(3)=5,  g(3)=8,  f(g(3))=10\boxed{f(3) = 5,\; g(3) = 8,\; f(g(3)) = 10}

Key Takeaways

  • f(x)f(x) is just notation for the output of function ff at input xx.
  • Substitute carefully—replace every xx with the given input.
  • Algebraic inputs like x+1x+1 or 2x2x require expanding and simplifying.
  • On SAT, evaluation may be numeric or symbolic—read carefully.
  • For composed or nested functions, compute inside-out.