21M.2.SL.TZ2.5

pestleMathematicsAASLPaper 221M· sl-2-2-functions-notation-domain-range-and-inverse-as-reflection, sl-2-5-composite-functions-identity-finding-inversesource ↗

The functions $f$ and $g$ are defined for $x \in ℝ$ by $f (x) = 6 x^{2} - 12 x + 1$ and $g (x) = - x + c$ , where $c \in ℝ$ .

Find the range of $f$ .

[2]

Given that $(g \circ f) (x) \leq 0$ for all $x \in ℝ$ , determine the set of possible values for $c$ .

[4]

Markscheme / solution

attempting to find the vertex (M1)

$x = 1$ OR $y = - 5$ OR $f (x) = 6 {(x - 1)}^{2} - 5$

range is $y \geq - 5$ A1

[2 marks]

METHOD 1

$(g \circ f) (x) = - (6 x^{2} - 12 x + 1) + c (= - (6 {(x - 1)}^{2} - 5) + c)$ (A1)

EITHER

relating to the range of $f$ OR attempting to find $g (- 5)$ (M1)

$5 + c \leq 0$ (A1)

attempting to find the discriminant of $(g \circ f) (x)$ (M1)

$144 + 24 (c - 1) \leq 0 (120 + 24 c \leq 0)$ (A1)

THEN

$c \leq - 5$ A1

METHOD 2

vertical reflection followed by vertical shift (M1)

new vertex is $(1, 5 + c)$ (A1)

$5 + c \leq 0$ (A1)

$c \leq - 5$ A1

[4 marks]

Examiners’ report

[N/A]