Questions and Answers
State the domain for which f(x) is defined.
The domain is the set of all real numbers for which f(x) produces a valid output.
Which inequality defines the domain of the function f(x)?
What values of x satisfy the equation f(x) = 0?
Values x such that f(x) = 0.
Which inequality gives the domain of the function f(x)?
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Find the domain of the function given in Functions, 2ADV F1 2017 HSC 11h.
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The mass of a crab is directly proportional to the cube of the diameter of its shell. If a crab has a shell diameter of 15 cm and weighs 680 grams, what will be the diameter of a crab that weighs 1.1 kilograms? Give your answer to 1 decimal place. The diameter is ______.
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What is the stopping distance of a car travelling at 105 km/h, if a car travelling at 70 km/h takes 58.8 meters to stop?
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What is the slope of the line with equation y = mx + c?
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What are the coordinates of the final image after the transformations?
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The equation of the new graph after transformations is?
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What are the values of parameters in the function?
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The graph has a vertical asymptote with equation ____.
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What transformations are applied to the function?
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Describe each transformation applied to the function.
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What are the coordinates of the horizontal axis intercepts?
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What is the final equation of the curve after transformations?
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What is the gradient of the line?
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Which equation represents the line perpendicular to the given line, passing through the point?
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What is the y-intercept of the line?
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Which pair of equations model the situation where Luke and Rhys gain and lose followers?
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How many calendars does Ita need to sell to break even?
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What is the total cost of purchasing and running four light globes for one year?
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How much profit will Fiona and John make if 80 people attend their event?
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How long will it take Ariana to repay the loan in full?
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What is the mean life, in hours, of the light globes if 97.5% will last up to 5000 hours?
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What is the direct variation equation relating British pounds to Australian dollars?
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Which expression is equal to ___?
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Which diagram shows the graph of an odd function?
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Factorise ___ (2 marks)
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Find the points of intersection of ___ and ___ (3 marks)
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Solve ___ (2 marks)
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Simplify ___ (1 mark)
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The graph of the function ___ has asymptotes at ___
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What are the asymptotes of ___?
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Identify the domain of ___ (1 mark)
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Find the values of ___ for which ___ is always positive (1 mark)
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What is the value of $x$ correct to two decimal places?
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What is $x$ correct to three significant figures?
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What is the value of $x$, correct to three significant figures?
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What is $x$ written in scientific notation, correct to four significant figures?
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Evaluate $x$ correct to four significant figures.
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Evaluate $x$ correct to three significant figures.
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Expand and simplify $x$.
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Solve $x$.
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Find $x$ and $y$ such that they are real numbers.
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Worker A picks a bucket of blueberries in ___ hours.
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Write an algebraic ______ for the fraction of a bucket of blueberries that could be picked in one hour if A and B worked together.
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Find the reciprocal of $x$.
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Simplify $x$ to its simplest form.
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Write down the rule of the function $f(x)$?
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If the equation $x = y$ is true for all real values of $x$, then $y$ could equal?
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Given $x$ and $y$, sketch $f(x)$ over its natural domain.
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Find the domain and range of function $f(x)$ given certain conditions.
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State the range of function $f(x)$.
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Let $f(x) = g(x)$ where $f$ and $g$ are functions. Show that $h(x) = k(x)$.
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Study Notes
Functions Overview
- F1 Working with Functions is an essential Year 11 topic in the new Advanced course.
- It covers 5 main categories: Algebraic Techniques, Linear Functions, Quadratics and Cubic Functions, Composite Functions, and Further Functions and Relations.
- Analysis indicates that functions contribute approximately 7.1% to exams based on historical data.
Common Assessment Strategies
- The practice includes multiple-choice questions (MCQs) focused on calculating values, rationalizing denominators, and evaluating expressions to specified significant figures.
- Rounding rules may require knowledge of standard decimal rounding, significant figures, and scientific notation.
Historical Context
- Algebraic Techniques traditionally assessed included surd calculations, particularly tricky compound surds, which are no longer directly examinable.
- Questions around algebraic fractions have recently gained attention, bridging standards between Advanced and Standard 2 content.
Composite Functions Specifics
- Composite Functions is a newly introduced topic with no prior exam history but is highly examinable.
- Students should focus on identifying the domain and range of composite functions heavily tested in assessments.
Exam Preparation Insights
- Expect to see questions regarding evaluating functions under conditions, such as values written in scientific notation or finding the domain/range.
- Practice evaluating expressions to various significant figures while being mindful of precision to avoid penalties for incorrect rounding.
Recommended Focus Areas
- Review surd calculations, simplification of algebraic expressions, and rationalization strategies.
- Familiarize with questions that incorporate composite functions, especially where students must apply the rules defining their behavior.
- Incorporate examples of past exams to gain insights into common pitfalls and frequently examined topics.
Analytical Data Points
- Analytics reveal a consistent contribution percentage across various exam papers, suggesting stability in key topic areas.
- Emphasize training in areas showing historical trends in student performance, particularly those showing mean marks below 50%.
Tips for Success
- Consistent practice with past exam questions will reinforce understanding and improve confidence.
- Collaborative study groups can help clarify complex concepts like composite functions and algebraic expressions.
- Always double-check work to ensure accuracy in calculations and adherence to specified precision levels.### Functions Overview
- F1 Working with Functions is a Year 11 topic covering both new content and re-categorized old syllabus content.
- The topic comprises five main categories:
- Algebraic Techniques
- Linear Functions
- Quadratics and Cubic Functions
- Composite Functions
- Further Functions and Relations
HSC Analysis
- Average contribution to Advanced HSC exams over the past decade is noted.
- Functions and inequalities based on absolute values have shifted from the old to the Ext1 course.
- Graphing absolute value equations is emphasized as a crucial skill expected to be frequently examined.
- Understanding linear and square root inequalities is essential; they have appeared in several past Advanced papers.
Problem-Solving Techniques
- Emphasis on carefully presenting working solutions, ensuring clarity in format and layout.
- Students should be aware of common pitfalls, particularly with negative signs and division in inequalities.
- Equalities involving circle functions and proportional relationships are likely to feature in assessments.
Graphing and Function Types
- Being able to find the domain and range of functions, including circles, is crucial.
- Sketched graphs should accurately reflect mathematical relationships and significant points such as asymptotes.
- Students should practice the implications of various changes to function graphs, such as reflections.
Practical Applications
- Real-world applications of functions are highlighted, such as electrical circuit current related to resistance, and biological measures like crab mass relative to shell diameter.
- The concept of direct proportionality is explored in scenarios involving mass, speed, and stopping distances.
Exam Preparation and Common Errors
- Mean marks indicate areas that require extra focus, with penalties often stemming from misunderstanding function types and inappropriate methods used.
- Review of previous HSC items is advised to strengthen problem-solving capabilities in real exam conditions.
- Focus on arithmetic accuracy, particularly in inequalities and algebraic manipulation, is crucial for achieving higher marks.### Breakeven Analysis
- Ita sells calendars for $25 each, with production costs at $8 each plus a fixed setup cost of $5950.
- Breakeven point calculations:
- Revenue from selling x calendars: 25x
- Total cost: 8x + 5950
- To breakeven, set revenue equal to total cost:
- 25x = 8x + 5950
- Solving gives x = 305 calendars.
Line Equation and Perpendicularity
- For a line through a given point and perpendicular to another line, determine the slope of the original line and use the negative reciprocal for the new line's slope.
- Utilize point-slope form: y - y1 = m(x - x1).
Graphing Functions
- Intercepts on both axes play a crucial role in sketching graphs: x-intercept occurs where y=0, and y-intercept where x=0.
- For given volume functions, understanding how to determine when two tanks contain the same volume aids in analytical graphing.
Value Depreciation
- Damon experiences depreciation on a pool pump at 8% of its original price each year.
- The function implementing this depreciation can be modeled as:
- V(t) = 3250(1 - 0.08)^t, where t is time in years.
Charity Event Costs and Revenue
- Fixed costs for the event: venue hire ($650) and band ($850), variable costs for drinks at $25 per person.
- Total cost function:
- C(n) = 1500 + 25n, where n is the number of attendees.
- Break-even analysis requires equating cost to revenue from ticket sales.
Tax and Income Relations
- Tax payable is determined by taxable income ranges; tax behavior changes significantly after certain thresholds.
- Graphical interpretation can help to estimate tax liabilities at specified income levels.
Light Globe Cost Analysis
- A clubhouse utilizes four globes for 5 hours nightly, costing $6 each and additional running costs per hour.
- Total cost equation for one year incorporates both purchase and operating expenses.
Fundraising Event Profitability
- Estimating optimal ticket pricing is essential for maximizing profit while ensuring attendance.
- Profit can be calculated by: Profit = Revenue from tickets - Total costs.
Capacity Constraints in Production
- A factory has production limits on boots and sandals, which must be expressed with linear inequalities.
- Maximums for both products inform practical limits in production scenarios.
General Graphing Techniques
- When encountering graph problems (odd/even functions, asymptotes), analysis of transformations and properties can guide constructions and predictions.
- Efficient use of equations represents various aspects of apples-to-oranges comparisons in real-world models.
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Description
Test your understanding of advanced functions in Algebra for Year 11. This quiz focuses on key concepts and techniques related to functions as taught in the F1 curriculum. Prepare to tackle multiple choice questions designed to challenge your knowledge and skills.
