Higher Maths: functions, graphs, exponentials and logarithms

Before the calculus, Higher Maths asks you to be fluent with functions themselves: combining them, reversing them, moving their graphs about, and handling the exponential and logarithmic pair that turns up everywhere from growth models to solving for an unknown exponent.

Ten single-answer questions cover composite and inverse functions, the translation described by y=f(x−3)+2, why a logarithm needs a strictly positive argument, the quotient law, solving a logarithmic equation, an exponential growth model evaluated at a given time, solving 5x=20 with logarithms, the range of a quadratic, and the reflection that relates any function to its inverse. Two of the wrong options are worth noticing: in the domain question and the range question the near-miss differs from the key only in whether the inequality is strict or inclusive, so it is wrong at exactly one boundary value — the distinction Higher expects you to make deliberately rather than by habit.

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  • Evaluate a composite function in the correct order
  • Find the inverse of a linear function by rearranging and swapping variables
  • Describe the translation given by a change inside and outside a function
  • State the domain of a logarithmic function, knowing the argument must be strictly positive
  • Apply the quotient law of logarithms and evaluate the result
  • Convert between logarithmic and exponential form to solve an equation
  • Evaluate an exponential growth model at a given time rather than treating growth as linear
  • Solve for an unknown exponent by taking logarithms of both sides
  • State the range of a quadratic from its minimum value
  • Relate the graph of a function to the graph of its inverse by reflection in the line y = x

Solve $5^x = 20$, giving the answer to two decimal places. — Take the natural logarithm of both sides: $x \ln 5 = \ln 20$, so $x = \frac{\ln 20}{\ln 5} \approx \frac{2.9957}{1.6094} \approx 1.86$.

Sample question

If $f(x) = 3x - 5$, what is the inverse function $f^{-1}(x)$?

See the answer

$\frac{x + 5}{3}$

Set $y = 3x - 5$. Rearrange for $x$: $y + 5 = 3x$, so $x = \frac{y + 5}{3}$. Thus $f^{-1}(x) = \frac{x + 5}{3}$. A common error is failing to change the sign of the constant.

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