Basic Operations on Functions
- Given two functions \( f(x) \) and \( g(x) \), you can form:
- \( (f+g)(x) = f(x) + g(x) \)
- \( (f-g)(x) = f(x) - g(x) \)
- \( (f \times g)(x) = f(x) \times g(x) \)
- \( (f \div g)(x) = \frac{f(x)}{g(x)} \), provided \( g(x) \neq 0 \)
- These are evaluated pointwise: pick an \( x \), compute \( f(x) \) and \( g(x) \), then operate.
Sample: \(f(x) = x+1\) (blue),
\(g(x) = 2-x\) (green)
Domains for Combined Functions
- For \( f+g, f-g, f\times g \): domain is intersection of domains of \( f \) and \( g \).
- For \( f \div g \): intersection minus any \( x \) where \( g(x) = 0 \).
- For composition \( (f \circ g)(x) = f(g(x)) \): domain is all \( x \) in domain of \( g \) such that \( g(x) \) is in domain of \( f \).
Graphing Combined Functions
- Choose key \( x \)-values (intercepts, turning points).
- For each \( x \), read \( f(x) \) and \( g(x) \) from their graphs, compute the result, and plot.
- Sketch a smooth curve consistent with the originals.
Example Table:
| \(x\) | \(f(x)\) | \(g(x)\) | \(f+g\) | \(f-g\) | \(f×g\) | \(f÷g\) |
|---|---|---|---|---|---|---|
| \(-1\) | \(2\) | \(1\) | \(3\) | \(1\) | \(2\) | \(2\) |
| \(0\) | \(1\) | \(0\) | \(1\) | \(1\) | \(0\) | undefined |
| \(1\) | \(2\) | \(3\) | \(5\) | \(-1\) | \(6\) | \(2/3\) |
Visual Example:
Sample graphs: \(f(x) = x+2\)
(blue), \(g(x) = 2-x\) (green), \(f+g\) (red)
Zeros, Intercepts, and Asymptotes
- Sum: Zeros where \( f(x) + g(x) = 0 \). y-intercept: \( f(0) + g(0) \).
- Difference: Zeros where \( f(x) = g(x) \). y-intercept: \( f(0) - g(0) \).
- Product: Zeros where either \( f(x) = 0 \) or \( g(x) = 0 \). y-intercept: \( f(0) \times g(0) \).
- Quotient: Zeros where \( f(x) = 0 \) and \( g(x) \neq 0 \). Vertical asymptotes where \( g(x) = 0 \) and numerator nonzero. Holes if both zero.
Sample: \(f(x) = x\) (blue), \(g(x)
= 2-x\) (green), \(f/g\) (red, vertical asymptote)
Evaluating Examples Numerically
- Given algebraic \( f \) and \( g \):
- \( f(x) = x^2 + 1,\ g(x) = 3x \)
- \( (f+g)(1) = 2 + 3 = 5 \)
- \( (f-g)(2) = 5 - 6 = -1 \)
- \( (f\times g)(0) = 1 \times 0 = 0 \)
- \( (f\div g)(1) = 2/3 \)
- Given graphs: read values at \( x \), then operate.
Decomposing h(x) into f(x) and g(x)
- For \( h(x) = f+g \): y-intercept splits, zeros may be from cancellation.
- For \( h(x) = f\times g \): zeros of h are zeros of f or g. Use zero locations to propose factors.
- For \( h(x) = f\div g \): vertical asymptotes where \( g(x) = 0 \), zeros where \( f(x) = 0 \).
- For \( h(x) = f \circ g \): look for domain restriction patterns, repeated shapes.
Composite Functions (f∘g and g∘f)
- Definition: \( (f \circ g)(x) = f(g(x)) \). Notation: \( f \circ g \) or \( f(g(x)) \).
- Domain: \( \{ x \in \text{Domain}(g) : g(x) \in \text{Domain}(f) \} \)
- Range: subset of range of f, depends on g's outputs.
Example: If \( f(x) = 2x+1,\ g(x) = x^2 \), then \( (f \circ g)(x) = 2x^2+1 \), \( (g
\circ f)(x) = (2x+1)^2 \).
Sample: \(g(x) = x²\) (green),
\(f∘g = 2x²+1\) (blue)
How to Compute f∘g(x) and g∘f(x)
- Given formulas: substitute and simplify.
- Given graphs: read \(g(x)\) at \(x\), then f at that output.
- Always check if the output of the inner function is in the domain of the outer.
Example: If \( f(x) = \sqrt{x} \), \( g(x) = x-3 \):
\( (f \circ g)(x) = \sqrt{x-3} \), domain: \( x \geq 3 \).
\( (g \circ f)(x) = \sqrt{x} - 3 \), domain: \( x \geq 0 \).
\( (f \circ g)(x) = \sqrt{x-3} \), domain: \( x \geq 3 \).
\( (g \circ f)(x) = \sqrt{x} - 3 \), domain: \( x \geq 0 \).
Sample: \(g(x) = x-3\) (green),
\(f∘g = \sqrt{x-3}\) (blue, only for \(x≥3\))
Evaluating Compositions from Graphs
- To find \( (f \circ g)(a) \):
- Read \( g(a) \) from g's graph.
- Read \( f(g(a)) \) from f's graph.
- If \( g(a) \) not in domain of f, undefined.
- To find \( (g \circ f)(b) \): start with f(b), then g at that value.
Example: If g(2) = -1, f(-1) = 4, then \( (f \circ g)(2) = 4 \).
Common Exam Pitfalls & Tips
- Don't confuse \( f(g(x)) \) with \( f(x) \cdot g(x) \).
- Domain of \( f \circ g \) requires checking both domain(g) and that \( g(x) \) lands in domain(f).
- For sums/differences, zeros are not just the union/intersection of zeros.
- For quotients, check for holes (removable discontinuities).
- When decomposing \(h(x)\), pick simple, interpretable f and g and justify by intercepts/zeros/asymptotes.
Quick Study Checklist
- Compute \( (f \pm g)(a), (f \times g)(a), (f \div g)(a) \) from formulas or graphs.
- Find domain of any combined function.
- Compute \( f(g(a)) \) and \( g(f(a)) \) from formulas and graphs.
- Identify zeros, \(y\)-intercept, and possible vertical asymptotes of combinations.
- Explain why a composition is undefined at a particular \(x\).
- Propose reasonable \(f\) and \(g\) when given h and the operation, and justify by zeros/asymptotes/intercepts.