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 , 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 $5^x = 20$ 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 is the strict inequality rather than the inclusive one, which is wrong at exactly one value of x — 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
Given $f(x) = 2x + 1$ and $g(x) = x^2$, what is the value of $f(g(3))$?
- $37$
- $19$
- $49$
- $13$
See the answer
$19$
First, evaluate $g(3) = 3^2 = 9$. Then, substitute this into $f(x)$, so $f(9) = 2(9) + 1 = 19$. If one calculates $g(f(3))$ instead, one gets $g(7) = 49$.
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