Evaluating a Linear Function

If $f(x) = 3x - 2$, find $f(4)$.

Identify what to substitute: $f(4)$ means we replace $x$ with $4$ = Input = 4

Substitute the value: $f(4) = 3(4) - 2$ = $3(4) - 2$

Multiply first: $3 \times 4 = 12$ = $12 - 2$

Subtract: $12 - 2 = 10$ = $f(4) = 10$

Answer: $f(4) = 10$

Evaluating with a Negative Input

If $g(x) = x^2 + 1$, find $g(-3)$.

Identify the substitution: $g(-3)$ means replace $x$ with $-3$ = Input = -3

Substitute the value: $g(-3) = (-3)^2 + 1$ = $(-3)^2 + 1$

Square the number: $(-3)^2 = (-3) \times (-3) = 9$ = $9 + 1$

Add: $9 + 1 = 10$ = $g(-3) = 10$

Answer: $g(-3) = 10$

Finding an Input from an Output

If $h(x) = 2x + 6$, and $h(a) = 14$, find $a$.

Set up the equation: If $h(a) = 14$, then $2a + 6 = 14$ = $2a + 6 = 14$

Subtract 6 from both sides: $2a + 6 - 6 = 14 - 6$ = $2a = 8$

Divide by 2: $\frac{2a}{2} = \frac{8}{2}$ = $a = 4$

Verify the answer: $h(4) = 2(4) + 6 = 8 + 6 = 14$ ✓ = $a = 4$

Answer: $a = 4$

Mistake: Thinking $f(x)$ means $f$ multiplied by $x$

Why: The notation $f(x)$ looks like multiplication, but it actually means "the function $f$ applied to the input $x$." The parentheses indicate a function, not multiplication.

Correct: Read $f(x)$ as "f of x" — it represents the output when $x$ is the input.

Mistake: Forgetting to substitute everywhere

Why: When $x$ appears multiple times in a function, all instances must be replaced.

Correct: For $f(x) = x^2 + 2x$, to find $f(3)$: replace BOTH $x$'s to get $3^2 + 2(3) = 9 + 6 = 15$.

Mistake: Confusing the function name with the variable

Why: Students sometimes mix up $f$ (the function name) with $x$ (the input variable).

Correct: $f$ names the rule; $x$ is what goes in. Different functions can use the same variable: $f(x) = 2x$ and $g(x) = x + 5$.

Temperature Conversion

Converting between Celsius and Fahrenheit uses function notation.

The function $F(C) = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit. So $F(25) = \frac{9}{5}(25) + 32 = 45 + 32 = 77°F$.

Ride-Share Pricing

Ride-share apps calculate fares using functions.

If the fare is $C(m) = 2.50 + 1.75m$ where $m$ is miles, then a 6-mile ride costs $C(6) = 2.50 + 1.75(6) = 2.50 + 10.50 = 13$ dollars.

Q: Why do we use letters like $f$ instead of just writing the equation?

A: Function names let us work with multiple relationships at once. Instead of saying "the first equation" and "the second equation," we can say $f(x)$ and $g(x)$. This is especially useful when comparing functions or combining them.

Q: Can any letter be used for a function name?

A: Yes! While $f$, $g$, and $h$ are most common, any letter works. In science, meaningful letters are often used: $v(t)$ for velocity as a function of time, $A(r)$ for area as a function of radius.

Q: Is $f(x) = y$ the same as $y = f(x)$?

A: Yes, they mean the same thing. $f(x)$ represents the output of the function, which we often call $y$. So $f(x) = 2x + 1$ is equivalent to $y = 2x + 1$ when working with the function $f$.