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Longhand

Chapter 0: The basics

Functions, graphs and inverses

Everything later in the course is applied to functions, so the term opens by making sure you can move between a formula, a graph and a description in words. You work with domain and range, compose and invert functions, predict what a shift, stretch or reflection does to a graph, and solve the exponential and trigonometric equations that later chapters take for granted.

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Practice questions

Every question below is original material written for Longhand. Open a solution to read the full working, or answer it yourself to have it counted towards your progress.

  • Introductory2 marks · 3 min

    Let f(x)=2x−5f(x) = 2x - 5 and g(x)=x2+1g(x) = x^{2} + 1. Evaluate the composition below.

    (f∘g)(3)(f \circ g)(3)
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  • Introductory1 mark · 2 min

    Find the largest value of xx in the domain of the function below.

    g(x)=7−2xx+3g(x) = \frac{\sqrt{7 - 2x}}{x + 3}
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  • Introductory2 marks · 3 min

    The graph of y=f(x)y = f(x) is shifted 3 units to the right and then reflected in the xx-axis. Which formula describes the resulting graph?

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  • Standard2 marks · 3 min

    Let f(x)=3x−1x+2f(x) = \dfrac{3x - 1}{x + 2}. Find f−1(1)f^{-1}(1).

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  • Standard2 marks · 4 min

    Find the sum of all real solutions of the equation below.

    22x−6⋅2x+8=02^{2x} - 6\cdot 2^{x} + 8 = 0
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  • Standard2 marks · 3 min

    Let h(x)=1+e2xh(x) = \sqrt{1 + e^{2x}}. Suppose h(x)=f(g(x))h(x) = f\bigl(g(x)\bigr) with the inner function g(x)=e2xg(x) = e^{2x}. Which formula gives the outer function ff?

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  • Standard3 marks · 5 min

    Find the sum of all solutions of the equation below in the interval [0,2π)[0, 2\pi).

    2sin⁡2x−sin⁡x−1=02\sin^{2} x - \sin x - 1 = 0
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  • Challenging4 marks · 7 min

    Let f(x)=ln⁡ ⁣(1+x1−x)f(x) = \ln\!\left(\dfrac{1 + x}{1 - x}\right). Find the domain of ff, show that ff is odd, and find a formula for f−1(y)f^{-1}(y).

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  • Challenging3 marks · 5 min

    Let f(x)=x3+xf(x) = x^{3} + x. Explain briefly why ff is one-to-one, then find f−1(10)f^{-1}(10) without finding a formula for f−1f^{-1}.

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  • Capstone5 marks · 8 min

    A function ff satisfies the identity below for every x≠2x \neq 2. Find f−1(3)f^{-1}(3).

    f(3x−1)=2x+5x−2f(3x - 1) = \frac{2x + 5}{x - 2}
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Related topics

  • Limits

    Limit laws, the indeterminate form 0/0, one-sided limits, the squeeze theorem and limits at infinity.

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