MA.912.F.3.2

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.2.4
• MA.912.AR.2.5 
• MA.912.AR.3.7
• MA.912.AR.3.8 
• MA.912.AR.4.4 
• MA.912.AR.5.6
• MA.912.AR.5.7
• MA.912.AR.5.8
• MA.912.AR.5.9 
• MA.912.F.1.1

Terms from the K-12 Glossary

• Domain 
• Function 
• Function notation

Vertical Alignment

Previous Benchmarks

• MA.7.AR.1 
• MA.8.AR.1 
• MA.912.AR.1.4
• MA.912.AR.1.7

Next Benchmarks

• MA.912.C.2.3
• MA.912.C.2.4

Purpose and Instructional Strategies

• When appropriate, the domain restrictions will be determined for the new function. Students will evaluate the solution when combining two functions for a provided input. 
• In this benchmark, students will combine functions through addition, subtraction, multiplication, and division. This process will utilize their prior experience with polynomial arithmetic in MA.912.AR.1 from Algebra I. 
  • In mathematical contexts, combinations through addition may be represented as ($f$ + $g$)($x$) or as $h$($x$) = $f$($x$) + $g$($x$). 
  • In mathematical contexts, combinations through subtraction should be represented as ($f$ − $g$)($x$) or as $h$($x$) = $f$($x$) − $g$($x$). 
  • In mathematical contexts, combinations through multiplication should be represented as ($f$ · $g$)($x$) or as $h$($x$) = $f$($x$) · $g$($x$). 
  • In mathematical contexts, combinations through division should be represented as ($\frac{\text{<math><mi>f</mi></math>}}{\text{<math><mi>g</mi></math>}}$) ($x$) or as $h$($x$) = $\frac{\text{<i>f(x)</i>}}{\text{<i>g(x)</i>}}$ where $g$($x$) ≠ 0. Additionally for division, it can be represented using the division symbol. 
• Explain that when functions are combined using addition, subtraction, and multiplication, the domain of the resulting function is only the inputs ($x$-values) that are common to the domains of the original functions. The domain of $f$ + $g$, $f$ − $g$ and $f$ · $g$ is the intersection of the domains of $f$ and $g$. 
• When combining functions by division, students will need to consider values in the domain of the quotient that should be restricted. Values that should be restricted are those that would cause the denominator to equal zero. 
• Instruction in Algebra I included representing domain, range and constraints using words, inequality notation and set-builder notation. In Math for College Algebra, instruction also includes interval notation. 
  • Words 
    • If the domain is all real numbers, it can be written as “all real numbers” or “any value of $x$, such that $x$ is a real number.” 
  • Inequality notation 
    • If the domain is all values of $x$ greater than 2, it can be represented as $x$ >2. 
  • Set-builder notation 
    • If the range is all values of $y$ less than or equal to zero, it can be represented as {$y$|$y$ ≤ 0} and is read as “all values of $y$ such that $y$ is less than or equal to zero.” 
  • Interval notation 
    • If the domain is all values of $x$ less than or equal to 3, it can be represented as (−∞, 3]. If the domain is all values of $x$ greater than 3, it can be represented as (3, ∞). If the range is all values greater than or equal to −1 but less than 5, it can be represented as [−1, 5). 
• Instruction may include the use of graphs and tables to foster a deeper understanding of operations with functions. 
• Students have previous knowledge of linear and quadratic functions, and should make the connection that by multiplying two linear functions, it results in a quadratic function. 
  • For example, in order to determine revenue, one could multiply the function that represents price of a product by the function that represents quantity of a product that will be sold at that price.

Common Misconceptions or Errors

• When multiplying and dividing functions students may struggle working with exponents, remind students of the multiplication and division properties of exponents, include negative exponents. 
• When finding the domain of the new function, students find the union of the domains instead of the intersection. Emphasize that they need to find what the domains have in common. 
• When finding the domain of $\frac{\text{<math><mi>f</mi></math>}}{\text{<math><mi>g</mi></math>}}$ students forget to identify the restrictions on the domain. Remind students that they always need to find the values that make $g$($x$) = 0 and exclude them from the domain of $\frac{\text{<math><mi>f</mi></math>}}{\text{<math><mi>g</mi></math>}}$.

Instructional Tasks

• Using the graphs below, sketch a graph of the function $s$($x$) = $f$($x$) + $g$($x$).
  
  
  

Instructional Items

• Given $f$($x$) = $x$² +2$x$ + 4 and $f$($x$) = −2$x$ + 5, find $f$($x$) + $g$($x$), $f$($x$) − $g$($x$), $f$($x$) · $g$($x$) and $\frac{\text{<i>f(x)</i>}}{\text{<i>g(x)</i>}}$. Determine the domain for each function.

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