Range and Why a Function Is Not One-One

This question chains three linked ideas the 0606 functions topic loves: reading a maximum from completed-square form, stating a range, and understanding why a parabola is not one-one (and how a domain restriction fixes that).

A function is defined by $f(x) = 5 - (x - 2)^2$ for $x \in \mathbb{R}$. (i) State the greatest value of $f(x)$ and the value of $x$ at which it occurs. [2] (ii) State the range of $f$. [1] (iii) Explain why $f$ does not have an inverse, and state the largest value of $k$ for which $f$ defined on $x \le k$ is one-one. [3]

The working

(i) The expression is already in completed-square form. Since $(x - 2)^2 \ge 0$, subtracting it from $5$ makes $f$ largest when the square is zero: $\text{greatest value} = 5, \text{ at } x = 2 \quad \text{(B1, B1)}$

(ii) Nothing can be added above $5$, and $f$ decreases without limit as $x$ moves away from $2$: $\text{range: } f(x) \le 5 \quad \text{(B1)}$

(iii) $f$ is a "$\cap$"-shaped parabola with vertex at $x = 2$. Every value below $5$ is produced by two $x$-values (one each side of $x = 2$), so $f$ is many-one and has no inverse over all reals. (B1)

To make it one-one, restrict the domain to one side of the vertex. The largest such interval ending at the vertex is $x \le 2$: $k = 2 \quad \text{(M1, A1)}$

Where the marks are won and lost

• Because $(x - 2)^2$ is subtracted, the vertex is a maximum, not a minimum. Reading it as a minimum flips the range the wrong way.
• The range is $f(x) \le 5$ (one-sided). Writing $f(x) \ge 5$ is the standard slip for a downward parabola.
• The one-one explanation must mention two inputs sharing an output (or the horizontal-line idea). “It’s a parabola” alone is not the reasoning the mark rewards.

Common mistakes

• Treating the vertex as a minimum and giving range $f(x) \ge 5$.
• Choosing $k$ on the wrong side, or an interval that still spans the vertex.
• Saying $f$ has no inverse “because it’s quadratic” without the many-one reason.

Full method: One-One Functions notes. See also Domain & Range and the Functions pillar.