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  1. Problems
  2. /Functions and Calculus

Functions and Calculus

Mathematics
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41 problems
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1.
A parabola fitted to stopping distances measured between 20 and 80 km/h is d = 0.006v² + 0.15v +…
High School (11–12) · Functions as Models of DataStemPy LabStemPy Lab
Medium
2.
The areas of five circles were measured as 3.18 cm², 12.49 cm², 28.11 cm², 50.39 cm² and 78.42 cm²…
High School (11–12) · Functions as Models of DataStemPy LabStemPy Lab
Medium
3.
A fixed amount of gas at a fixed temperature has a volume of 8 L at a pressure of 120 kPa.
High School (11–12) · Functions as Models of DataStemPy LabStemPy Lab
Easy
4.
A taxi charges Rs 60 for 3 km and Rs 105 for 6 km.
High School (11–12) · Functions as Models of DataStemPy LabStemPy Lab
Medium
5.
Let f(x) = x - 4 and g(x) = 2x. Find (g ° f)⁻¹, and check that it equals f⁻¹ ° g⁻¹ but not g⁻¹ °…
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Medium
6.
A has 3 elements and B has 6 elements. How many functions are there from A to B?
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Medium
7.
f(x) = -x² + 9 is a function from ℝ to ℝ. Is it onto? Change the codomain so that it becomes onto.
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Easy
8.
Show that f(x) = 4 - 3x is one-one. Show that g(x) = x² + 6x is not one-one on ℝ.
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Medium
9.
Find the inverse of f(x) = (2x + 5)/(x - 3). Give the domain and the range of f.
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Medium
10.
Write h(x) = 1/((3x - 1)²) as a composite of three simple functions.
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Easy
11.
Let f(x) = 2x + 3 and g(x) = ax + b. For which a and b is g ° f = f ° g?
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Hard
12.
Let f(x) = 1/(x + 2) and g(x) = √x. Find (g ° f)(x) and (f ° g)(x), with their domains.
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Hard
13.
Let f(x) = √(x + 5) and g(x) = x² - 4. Find the domain of f + g and the domain of f/g.
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Medium
14.
Let f(x) = x + 5 and g(x) = 4x - 1. Find (g ° f)(x) and (f ° g)(x), and their values at x = 3.
High School (11–12) · Composite and Inverse FunctionsStemPy LabStemPy Lab
Easy
15.
Write f(x) = (x - 2)³ as the sum of an even function and an odd function.
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Hard
16.
Show that {x} = x - [x] has period 1. Then find the period of {5x}.
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Medium
17.
For f(x) = -x² + 10x, find the interval on which f is increasing, the interval on which it is…
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Medium
18.
Say whether each function is even, odd or neither. 1/x³; x⁴ + x²; x² + x; x|x|
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Easy
19.
Describe the graph of y = 1/(x + 1) - 2. Give the asymptotes, the domain and the range.
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Medium
20.
Describe how to get the graph of y = √(9 - x) from the graph of y = √x.
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Hard
21.
The point (3, 4) is on the graph of y = f(x).
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Medium
22.
Describe how to get the graph of y = -3(x + 2)² + 4 from the graph of y = x².
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Medium
23.
Write f(x) = |x + 2| + |x - 3| without bars for (a) x < -2, (b) -2 ≤ x ≤ 3 and (c) x > 3.
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Hard
24.
Solve (a) |3x + 1| < 8 and (b) |x - 2| ≥ 5.
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Medium
25.
A function is defined by f(x) = 2x + 1, x < 0; x², 0 ≤ x ≤ 3; 12 - x, x > 3.
High School (11–12) · Graphs of FunctionsStemPy LabStemPy Lab
Medium
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