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Introduction to Algebra / LEVEL 3 · DIFFICULTY 3/5

Functions and Reversible Rules

Compose, reverse and constrain algebraic rules.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST ALG 3.1Function CompositionRead the lesson
MINI QUEST ALG 3.2Inverse FunctionsRead the lesson
MINI QUEST ALG 3.3Linear InequalitiesRead the lesson
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STAGE ALG 3.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Function Composition

Useful preparation: Completing the Square

Goal: Follow two rules in their specified order.

Before you begin: Completing the Square

Understand the idea

A function assigns one output to each allowed input. Composition feeds one function’s output into another. The inside notation describes the first step.

(f∘g)

\displaystyle \left(x\right) = f\, \left(g\, \left(x\right)\right)

Choose and carry out a method

Evaluate the inner expression fully, then use that result as the outer input. For symbolic composition, replace every occurrence of the outer variable with parentheses containing the inner rule.

Check the reasoning

Track domain restrictions through both steps. Even if the original input is allowed by the inner function, its output must also be allowed by the outer function.

WORKED EXAMPLE 1

Let $\displaystyle f\, \left(x\right) = 2\, x + 3$ and $\displaystyle g\, \left(x\right) = x^{2} - 6$ . Find f(g(2)).

Evaluate the inner function first; order matters.
$\displaystyle g\, \left(2\right) = 4 - 6 = -2$ ; $\displaystyle f\, \left(-2\right) = 2\, \left(-2\right) + 3$ .
The result is -1. Reversing the two steps would compute a different composition.

-1

WORKED EXAMPLE 2

Let $\displaystyle f\, \left(x\right) = 2\, x + 4$ and $\displaystyle g\, \left(x\right) = x^{2} - 7$ . Find f(g(3)).

Evaluate the inner function first; order matters.
$\displaystyle g\, \left(3\right) = 9 - 7 = 2$ ; $\displaystyle f\, \left(2\right) = 2\, \left(2\right) + 4$ .
The result is 8. Reversing the two steps would compute a different composition.

Common pitfalls

Possible mix-up: f(g(x)) means f(x) times g(x).

Composition passes an output into a new rule.

Possible mix-up: Reversing the rules always gives the same output.

Check both orders; composition usually is not commutative.

Explain it to yourself

Invent two simple rules whose order changes the answer.

Preview the eight practice prompts

Problem 1
Let $\displaystyle f\, \left(x\right) = 2\, x + 6$ and $\displaystyle g\, \left(x\right) = x^{2} - 9$ . Find f(g(5)).
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Problem 2
Let $\displaystyle f\, \left(x\right) = 2\, x + 7$ and $\displaystyle g\, \left(x\right) = x^{2} - 5$ . Find f(g(6)).
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Problem 3
Let $\displaystyle f\, \left(x\right) = 2\, x + 8$ and $\displaystyle g\, \left(x\right) = x^{2} - 6$ . Find f(g(7)).
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Problem 4
Let $\displaystyle f\, \left(x\right) = 2\, x + 9$ and $\displaystyle g\, \left(x\right) = x^{2} - 7$ . Find f(g(8)).
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Problem 5
Let $\displaystyle f\, \left(x\right) = 2\, x + 10$ and $\displaystyle g\, \left(x\right) = x^{2} - 8$ . Find f(g(9)).
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Problem 6
Let $\displaystyle f\, \left(x\right) = 2\, x + 11$ and $\displaystyle g\, \left(x\right) = x^{2} - 9$ . Find f(g(10)).
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Problem 7New context
A two-step device first maps input x to $\displaystyle x^{2} - 5$ , then doubles that output and adds 12. What final output results from input 11?
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Problem 8New context
A two-step device first maps input x to $\displaystyle x^{2} - 6$ , then doubles that output and adds 13. What final output results from input 12?
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Open stage ALG 3.1 in the student workspace →

STAGE ALG 3.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Inverse Functions

Useful preparation: Function Composition

Goal: Recover the input of a one-to-one linear function.

Before you begin: Function Composition

Understand the idea

An inverse reverses a function. It exists as a function when each output comes from exactly one allowed input. A nonconstant linear rule is one-to-one on the real numbers.

\displaystyle f\, \left(x\right) = ax + b \Rightarrow f^{-1}\, \left(y\right) = \frac{y - b}{a}

\displaystyle a \ne 0

Choose and carry out a method

Write $\displaystyle y = f\, \left(x\right)$ , solve for x in terms of y, and then rename the input variable. Reverse the order of operations as well as the operations themselves.

Check the reasoning

Apply the original function to your inverse result. You should recover the starting output. The superscript -1 in inverse notation does not mean reciprocal.

WORKED EXAMPLE 1

If $\displaystyle f\, \left(x\right) = 3\, x - 3$ , find $\displaystyle f^{-1}\, \left(12\right)$ .

An inverse recovers the input from the output.
Solve $\displaystyle 3\, x - 3 = 12$ , so $\displaystyle 3\, x = 15$ .
The original input is 5; this is not the reciprocal of the output.

WORKED EXAMPLE 2

If $\displaystyle f\, \left(x\right) = 3\, x - 4$ , find $\displaystyle f^{-1}\, \left(14\right)$ .

An inverse recovers the input from the output.
Solve $\displaystyle 3\, x - 4 = 14$ , so $\displaystyle 3\, x = 18$ .
The original input is 6; this is not the reciprocal of the output.

Common pitfalls

Possible mix-up: The inverse is $\displaystyle \frac{1}{f}\, \left(x\right)$ .

A reciprocal changes an output; an inverse recovers an input.

Possible mix-up: Every function has an inverse on its full domain.

Different inputs producing the same output prevent a single-valued inverse.

Explain it to yourself

Why does $\displaystyle x^{2}$ need a restricted domain before its inverse can be a function?

Preview the eight practice prompts

Problem 1
If $\displaystyle f\, \left(x\right) = 3\, x - 6$ , find $\displaystyle f^{-1}\, \left(18\right)$ .
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Problem 2
If $\displaystyle f\, \left(x\right) = 3\, x - 7$ , find $\displaystyle f^{-1}\, \left(20\right)$ .
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Problem 3
If $\displaystyle f\, \left(x\right) = 3\, x - 8$ , find $\displaystyle f^{-1}\, \left(22\right)$ .
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Problem 4
If $\displaystyle f\, \left(x\right) = 3\, x - 9$ , find $\displaystyle f^{-1}\, \left(24\right)$ .
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Problem 5
If $\displaystyle f\, \left(x\right) = 3\, x - 10$ , find $\displaystyle f^{-1}\, \left(26\right)$ .
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Problem 6
If $\displaystyle f\, \left(x\right) = 3\, x - 11$ , find $\displaystyle f^{-1}\, \left(28\right)$ .
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Problem 7New context
A calibration changes raw reading x to $\displaystyle 3\, x - 12$ . The calibrated reading is 30. What was the raw reading?
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Problem 8New context
A calibration changes raw reading x to $\displaystyle 3\, x - 13$ . The calibrated reading is 32. What was the raw reading?
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Open stage ALG 3.2 in the student workspace →

STAGE ALG 3.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Linear Inequalities

Useful preparation: Inverse Functions

Goal: Handle negative scaling and integer boundaries.

Before you begin: Inverse Functions

Understand the idea

An inequality describes a set of inputs. Adding the same amount preserves order. Multiplying by a negative number reverses order because it reflects the number line.

\displaystyle a < b \Rightarrow -a > -b

Choose and carry out a method

Isolate the unknown as in an equation, but reverse the comparison when dividing or multiplying by a negative value. Then translate a strict real bound into any requested integer bound.

Check the reasoning

Check a point inside the proposed set and its boundary. A strict inequality excludes equality; an inclusive inequality includes it.

WORKED EXAMPLE 1

What is the greatest integer x satisfying $\displaystyle -3\, x > -12$ ?

Dividing an inequality by a negative number reverses its direction.
$\displaystyle -3\, x > -12$ becomes $\displaystyle x < 4$ .
The greatest allowed integer is 3; $\displaystyle x = 4$ fails the strict inequality.

WORKED EXAMPLE 2

What is the greatest integer x satisfying $\displaystyle -3\, x > -15$ ?

Dividing an inequality by a negative number reverses its direction.
$\displaystyle -3\, x > -15$ becomes $\displaystyle x < 5$ .
The greatest allowed integer is 4; $\displaystyle x = 5$ fails the strict inequality.

Common pitfalls

Possible mix-up: Division never changes the inequality symbol.

Division by a negative reverses it.

Possible mix-up: The boundary of $\displaystyle x < k$ is an allowed value.

Strict inequalities exclude the boundary.

Explain it to yourself

Use a number-line example to explain why negative multiplication reverses order.

Preview the eight practice prompts

Problem 1
What is the greatest integer x satisfying $\displaystyle -3\, x > -21$ ?
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Problem 2
What is the greatest integer x satisfying $\displaystyle -3\, x > -24$ ?
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Problem 3
What is the greatest integer x satisfying $\displaystyle -3\, x > -27$ ?
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Problem 4
What is the greatest integer x satisfying $\displaystyle -3\, x > -30$ ?
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Problem 5
What is the greatest integer x satisfying $\displaystyle -3\, x > -33$ ?
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Problem 6
What is the greatest integer x satisfying $\displaystyle -3\, x > -36$ ?
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Problem 7New context
A game adjustment gives a change of -3x points. It must exceed -39 points. What is the greatest integer setting x allowed?
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Problem 8New context
A game adjustment gives a change of -3x points. It must exceed -42 points. What is the greatest integer setting x allowed?
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Open stage ALG 3.3 in the student workspace →

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Functions and Reversible Rules

Function Composition

Understand the idea

Choose and carry out a method

Check the reasoning

Let f (x)=2 x+3\displaystyle f\, \left(x\right) = 2\, x + 3f(x)=2x+3 and g (x)=x2−6\displaystyle g\, \left(x\right) = x^{2} - 6g(x)=x2−6. Find f(g(2)).

Let f (x)=2 x+4\displaystyle f\, \left(x\right) = 2\, x + 4f(x)=2x+4 and g (x)=x2−7\displaystyle g\, \left(x\right) = x^{2} - 7g(x)=x2−7. Find f(g(3)).

Common pitfalls

Explain it to yourself

Inverse Functions

Understand the idea

Choose and carry out a method

Check the reasoning

If f (x)=3 x−3\displaystyle f\, \left(x\right) = 3\, x - 3f(x)=3x−3, find f−1 (12)\displaystyle f^{-1}\, \left(12\right)f−1(12).

If f (x)=3 x−4\displaystyle f\, \left(x\right) = 3\, x - 4f(x)=3x−4, find f−1 (14)\displaystyle f^{-1}\, \left(14\right)f−1(14).

Common pitfalls

Explain it to yourself

Linear Inequalities

Understand the idea

Choose and carry out a method

Check the reasoning

What is the greatest integer x satisfying −3 x>−12\displaystyle -3\, x > -12−3x>−12?

What is the greatest integer x satisfying −3 x>−15\displaystyle -3\, x > -15−3x>−15?

Common pitfalls

Explain it to yourself

Let $\displaystyle f\, \left(x\right) = 2\, x + 3$ and $\displaystyle g\, \left(x\right) = x^{2} - 6$ . Find f(g(2)).

Let $\displaystyle f\, \left(x\right) = 2\, x + 4$ and $\displaystyle g\, \left(x\right) = x^{2} - 7$ . Find f(g(3)).

If $\displaystyle f\, \left(x\right) = 3\, x - 3$ , find $\displaystyle f^{-1}\, \left(12\right)$ .

If $\displaystyle f\, \left(x\right) = 3\, x - 4$ , find $\displaystyle f^{-1}\, \left(14\right)$ .

What is the greatest integer x satisfying $\displaystyle -3\, x > -12$ ?

What is the greatest integer x satisfying $\displaystyle -3\, x > -15$ ?