This algebra 1 building functions worksheet gives 23 problems on building functions and inverses, ordered from easy to hard, with a separate answer key.
These 23 problems practice standards F-BF.A.1a, F-BF.A.1b, F-BF.B.4a and F-LE.A.1b. Start by writing functions from steps and recursive rules, then combine functions with sums, differences and products.
Next, solve f(x) = c and write inverses of linear functions. Finish with word problems about profit, taxi fares, savings and gas tanks that mix several skills.
Algebra 1 Building Functions Worksheet: problems 1–5
Problem 1: Steps to a function
Write a function f for this rule: multiply x by 4, then subtract 7. Then find \(f(5)\).
Problem 2: Recursive growth
Ethan’s bean plant is 6 inches tall in week 0 and grows 2 inches each week.
Write a recursive rule for the height \(h(n)\) after \(n\) weeks, then find \(h(4)\).
Problem 3: Recursive to explicit
A sequence is defined by \(f(0)=5\) and \(f(n)=f(n-1)+3\). Write an explicit rule for \(f(n)\) and find \(f(10)\).
Problem 4: Order of the steps
Write a function g for the steps shown, then find \(g(4)\).
Problem 5: Taxi fare function
A taxi charges a $3 flat fee plus $2.25 per mile. Write a function \(c(m)\) for the cost of a ride of \(m\) miles, then find the cost of an 8-mile ride.
Algebra 1 Building Functions Worksheet: problems 6–10
Problem 6: Sum of two functions
Let \(f(x)=3x+2\) and \(g(x)=x-5\). Write \((f+g)(x)\) and find \((f+g)(2)\).
Problem 7: Difference of two functions
Let \(f(x)=5x-1\) and \(g(x)=2x+8\). Write \((f-g)(x)\) and find \((f-g)(6)\).
Problem 8: Product of two functions
Let \(f(x)=x+3\) and \(g(x)=2x\). Write \((f\cdot g)(x)\) and find \((f\cdot g)(4)\).
Problem 9: Profit function
A club sells scarves. Revenue is \(R(x)=6x\) dollars and cost is \(C(x)=2x+40\) dollars for \(x\) scarves. Write the profit function \(P(x)\) and find the profit on 25 scarves.
Problem 10: Break-even from a graph
The school store sells hats. The graph shows revenue \(R(x)=7x\) and cost \(C(x)=3.5x+84\) in dollars. Use the equations to find how many hats must be sold to break even, and the revenue at that point.
Algebra 1 Building Functions Worksheet: problems 11–15
Problem 11: Solve f(x) = c
For \(f(x)=3x-4\), solve \(f(x)=20\) for \(x\).
Problem 12: Inverse of a linear function
Let \(f(x)=5x+10\). Solve \(y=5x+10\) for \(x\) to write an expression for the inverse.
Problem 13: Inverse with a fraction
Let \(f(x)=\frac{x}{4}+6\). Write an expression for the inverse, then use it to find the input that gives \(f(x)=9\).
Problem 14: Check an inverse
Let \(f(x)=3x-7\) and \(g(x)=\frac{x+7}{3}\). Find \(g(f(5))\) and \(f(g(5))\). What do the results show about \(g\)?
Problem 15: Is the rate constant?
Decide whether the table shows a constant rate of change. If it does, state the rate per 1 unit of \(x\) and write a function.
Algebra 1 Building Functions Worksheet: problems 16–20
Problem 16: Compare two tables
Which table changes at a constant rate per unit interval? Explain with the changes in \(y\).
Problem 17: Temperature conversion
Fahrenheit temperature is \(F(c)=1.8c+32\) for a Celsius temperature \(c\). (a) Find the Celsius temperature when it is 68°F. (b) Write the inverse expression for \(c\) in terms of \(F\).
Problem 18: Play ticket profit
Aisha sells play tickets for $8 each. Costs are $150 for the set plus $1.25 per ticket for printing. Write \(R(x)\), \(C(x)\) and \(P(x)\), then find the fewest tickets that give a profit.
Problem 19: Taxi fare inverse
The taxi fare is \(c(m)=2.25m+3\) dollars for \(m\) miles. Write \(m\) as a function of the cost \(c\), then find how many miles Priya rode for a fare of $25.50.
Problem 20: Savings recursion
Noah has $45 saved and adds $12 each week. Write a recursive rule and an explicit rule for the savings \(S(n)\) after \(n\) weeks, then find after how many weeks he has $165.
Algebra 1 Building Functions Worksheet: problems 21–23
Problem 21: Hot dog stand profit
A Little League stand sells hot dogs for $3.50 each. Each hot dog costs $1.10 to make, and the cart rental is $48. Write \(P(x)\), find the break-even number of hot dogs, and find the profit for 50 hot dogs.
Problem 22: Gas tank inverse
Diego starts a road trip with 14 gallons in the tank. The car uses 0.04 gallon per mile, so the gas left is \(g(m)=14-0.04m\). Write \(m\) in terms of \(g\), then find the miles driven when 6 gallons are left and when the tank is empty.
Problem 23: Two classes combined
After \(x\) hours of a fundraiser, Ms. Lee’s class has earned \(L(x)=4x-20\) dollars and Mr. Cho’s class has earned \(C(x)=6x-35\) dollars. (a) Write the total \(T(x)\). (b) After how many hours is the total $145? (c) Write \(x\) in terms of \(T\).
How to use this algebra 1 building functions worksheet
Print this algebra 1 building functions worksheet as a PDF, solve every problem on paper, then compare with the answer key.
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