Task Sets: Combine function types - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

1.

If $f(x) = x^2$ and $g(x) = 3x$ , what is $(f + g)(2)$ ?

2.

If $f(5) = 12$ and $g(5) = 3$ , what is $(f - g)(5)$ ?

A.

9

B.

15

C.

36

D.

4

3.

If $f(x) = 2x + 1$ and $g(x) = x^2$ , what is $(f \cdot g)(3)$ ?

B

Fluency Practice

1.

Let $f(x) = 2x + 1$ and $g(x) = x^2$ . Which expression equals $(f + g)(x)$ ?

A.

$x^2 + 2x + 1$

B.

$2x^2 + 1$

C.

$(2x + 1)(x^2)$

D.

$2x^2 + 1$ (from substituting $g$ into $f$ )

2.

Let $f(x) = 3x - 2$ and $g(x) = x + 5$ . Simplify $(f - g)(x) = ax + b$ . What is the coefficient $a$ ?

3.

Let $f(x) = x + 4$ and $g(x) = x - 1$ . What is the constant term of $(f \cdot g)(x)$ ?

4.

Let $f(x) = \sqrt{x}$ (domain: $x \geq 0$ ) and $g(x) = x + 3$ (domain: all reals). What is the domain of $(f + g)(x)$ ?

A.

All real numbers

B.

$x \geq 0$

C.

$x \geq -3$

D.

All reals except $x = 0$

5.

Let $f(x) = x^2 + 1$ and $g(x) = x - 2$ , both with domain: all reals. What additional restriction applies to the domain of $(f/g)(x)$ ?

A.

$x \neq -1$

B.

$x \neq 0$

C.

$x \neq 2$

D.

No additional restriction

C

Varied Practice

1.

A temperature model is $T(t) = 68 + 132(0.95)^t$ , where $t$ is in minutes. Identify the two component functions and explain what each represents physically.

2.

Let $f(x) = 5x$ and $g(x) = 3x - 2$ . Complete: $(f + g)(x) = \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}x + \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ .

coefficient of x:

constant:

3.

Let $f(x) = 4x + 3$ and $g(x) = 2x - 5$ . What type of function is $(f + g)(x)$ ?

A.

Quadratic — combining two linear functions creates a quadratic

B.

Linear — the sum of two linear functions remains linear

C.

Exponential — combining raises the degree

D.

The result has no standard type

4.

Let $f(x) = \sqrt{x+1}$ (domain: $x \geq -1$ ) and $g(x) = \frac{1}{x-4}$ (domain: $x \neq 4$ ). The domain of $(f + g)(x)$ is $x \geq -1$ and $x \neq$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

excluded value:

5.

A projectile's height is $h(t) = -16t^2 + 64t + 5$ feet. What is the height when $t = 2$ seconds?

D

Word Problems

1.

A coffee cup starts at 200°F in a 68°F room. The temperature follows $T(t) = 68 + 132(0.95)^t$ where $t$ is in minutes.

How many degrees above room temperature is the coffee after 10 minutes? Round to the nearest degree.

2.

A small business sells handmade items. Revenue is $R(x) = 50x$ dollars and cost is $C(x) = 200 + 20x$ dollars, where $x$ is the number of items sold.

1.

Write the profit function $P(x) = R(x) - C(x)$ , then evaluate the profit when 15 items are sold.

2.

How many items must be sold to break even (profit = 0)? Round up to the nearest whole item.

3.

A manufacturer's total cost is $T(q) = 0.5q^2 + 200$ dollars, combining fixed cost $F = 200$ and variable cost $V(q) = 0.5q^2$ .

Evaluate the variable cost $V(q)$ when $q = 20$ units.

E

Error Analysis

1.

A student writes: "If $f(x) = 2x$ and $g(x) = 3$ , then $(f + g)(x) = f(x + g(x)) = f(x + 3) = 2(x + 3) = 2x + 6$ ."

What error did the student make?

A.

The student computed composition $f(g(x))$ instead of the sum $f(x) + g(x)$

B.

The student used the wrong formula for $f$

C.

The algebra is correct — $2x + 6$ is the right answer

D.

The student should have computed $g(f(x))$ , not $f(g(x))$

2.

A student writes: "Revenue is $R(x) = 25x$ and cost is $C(x) = 500 + 10x$ . Profit equals revenue plus cost: $P(x) = 25x + 500 + 10x = 35x + 500$ ."

What error did the student make?

A.

The student used the wrong revenue formula

B.

Profit equals revenue minus cost, not revenue plus cost

C.

The algebra is wrong — $25x + 500 + 10x$ does not simplify to $35x + 500$

D.

Cost should not include a fixed component

F

Challenge / Extension

1.

The projectile model $h(t) = -16t^2 + 64t + 5$ combines three additive components. Identify each component and explain what physical phenomenon each represents.

2.