Evaluating Function Notation

Given $f(x) = x^2 - 3x + 2$, find $f(4)$ and $f(-1)$.

Find $f(4)$ by substituting $x = 4$: $f(4) = (4)^2 - 3(4) + 2$ = $f(4) = 16 - 12 + 2 = 6$

Find $f(-1)$ by substituting $x = -1$: $f(-1) = (-1)^2 - 3(-1) + 2$ = $f(-1) = 1 + 3 + 2 = 6$

Verify the calculations: Both inputs give output 6, which is valid = $f(4) = 6$ and $f(-1) = 6$

Answer: $f(4) = 6$ and $f(-1) = 6$. Note: Different inputs can produce the same output in a function.

Finding Domain and Range

Find the domain and range of $f(x) = \sqrt{x - 2}$.

Identify restrictions on domain: The expression under the square root must be non-negative: $x - 2 \geq 0$ = $x \geq 2$

Write domain in interval notation: All $x$ values from 2 to infinity = Domain: $[2, \infty)$

Determine possible outputs: Square root always gives non-negative results = $f(x) \geq 0$

Write range in interval notation: All non-negative real numbers = Range: $[0, \infty)$

Answer: Domain: $[2, \infty)$, Range: $[0, \infty)$

Function Operations

Given $f(x) = x + 3$ and $g(x) = x^2$, find $(f + g)(2)$ and $(f \cdot g)(2)$.

Understand the operations: $(f + g)(x) = f(x) + g(x)$ and $(f \cdot g)(x) = f(x) \cdot g(x)$ = Add or multiply the function outputs

Calculate $f(2)$ and $g(2)$: $f(2) = 2 + 3 = 5$ and $g(2) = 2^2 = 4$ = $f(2) = 5$, $g(2) = 4$

Find $(f + g)(2)$: $(f + g)(2) = f(2) + g(2) = 5 + 4$ = $(f + g)(2) = 9$

Find $(f \cdot g)(2)$: $(f \cdot g)(2) = f(2) \cdot g(2) = 5 \cdot 4$ = $(f \cdot g)(2) = 20$

Answer: $(f + g)(2) = 9$ and $(f \cdot g)(2) = 20$

Function Composition

Given $f(x) = 2x + 1$ and $g(x) = x^2$, find $(f \circ g)(3)$ and $(g \circ f)(3)$.

Understand composition notation: $(f \circ g)(x) = f(g(x))$ means apply $g$ first, then $f$ = Work from inside out

Find $(f \circ g)(3)$: First: $g(3) = 3^2 = 9$. Then: $f(9) = 2(9) + 1 = 19$ = $(f \circ g)(3) = 19$

Find $(g \circ f)(3)$: First: $f(3) = 2(3) + 1 = 7$. Then: $g(7) = 7^2 = 49$ = $(g \circ f)(3) = 49$

Compare the results: $19 \neq 49$ = Order matters! $(f \circ g) \neq (g \circ f)$

Answer: $(f \circ g)(3) = 19$ and $(g \circ f)(3) = 49$. Composition is not commutative.

Piecewise Functions

Evaluate the piecewise function at $x = -2$, $x = 0$, and $x = 3$: $$f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } x \geq 0 \end{cases}$$

Evaluate $f(-2)$: Since $-2 < 0$, use the first piece: $f(-2) = (-2)^2 = 4$ = $f(-2) = 4$

Evaluate $f(0)$: Since $0 \geq 0$, use the second piece: $f(0) = 2(0) + 1 = 1$ = $f(0) = 1$

Evaluate $f(3)$: Since $3 \geq 0$, use the second piece: $f(3) = 2(3) + 1 = 7$ = $f(3) = 7$

Summarize results: Each input falls into exactly one piece = $f(-2) = 4$, $f(0) = 1$, $f(3) = 7$

Answer: $f(-2) = 4$, $f(0) = 1$, $f(3) = 7$

Mistake: Confusing $f(x + 2)$ with $f(x) + 2$

Why: $f(x + 2)$ means substitute $(x + 2)$ into the function, while $f(x) + 2$ means add 2 to the output.

Correct: For $f(x) = x^2$: $f(x + 2) = (x + 2)^2 = x^2 + 4x + 4$, but $f(x) + 2 = x^2 + 2$. Very different!

Mistake: Reversing the order of composition: thinking $(f \circ g)(x) = g(f(x))$

Why: The notation can be confusing. The function on the right is applied first.

Correct: $(f \circ g)(x) = f(g(x))$: Apply $g$ first, then apply $f$ to the result.

Mistake: Forgetting to check domain restrictions when finding range

Why: The range depends on which inputs are allowed. A restricted domain often restricts the range too.

Correct: For $f(x) = x^2$ with domain $[0, 3]$, the range is $[0, 9]$, not $(-\infty, \infty)$.

Mistake: Assuming all relations are functions

Why: Not every equation or graph represents a function. The vertical line test must pass.

Correct: $x^2 + y^2 = 4$ (a circle) is NOT a function because vertical lines can intersect it twice.

Temperature Conversion

Converting between Celsius and Fahrenheit is a function composition application.

If $C(f) = \frac{5}{9}(f - 32)$ converts Fahrenheit to Celsius, then $C(68) = \frac{5}{9}(68 - 32) = \frac{5}{9}(36) = 20$ degrees Celsius.

Shipping Cost Calculator

Many businesses use piecewise functions to calculate shipping costs based on weight.

A company charges based on weight: 5 dollars for packages up to 1 kg, then 3 dollars per additional kg.

Q: Can a function have the same output for different inputs?

A: Yes! For example, $f(x) = x^2$ gives $f(2) = f(-2) = 4$. The rule is that each input has ONE output, but different inputs can share the same output.

Q: What's the difference between $f(x)$ and $f$ ?

A: $f$ is the function itself (a rule or process), while $f(x)$ is the output value when you input $x$. Think of $f$ as a machine and $f(x)$ as what comes out when you put $x$ in.

Q: Why does order matter in composition?

A: Because you're applying transformations in sequence. If $f$ doubles and $g$ adds 3: $f(g(2)) = f(5) = 10$, but $g(f(2)) = g(4) = 7$. Different processes yield different results.