Sketch the graphs of the functions $f(x) = x - 1$ and $g(x) = (1 - x)(3 + x)$ showing the $x$-intercepts - HSC - SSCE Mathematics Advanced - Question 19 - 2023 - Paper 1
Question 19
Sketch the graphs of the functions $f(x) = x - 1$ and $g(x) = (1 - x)(3 + x)$ showing the $x$-intercepts.
Hence, or otherwise, solve the inequality $x - 1 < (1 - x)... show full transcript
Worked Solution & Example Answer:Sketch the graphs of the functions $f(x) = x - 1$ and $g(x) = (1 - x)(3 + x)$ showing the $x$-intercepts - HSC - SSCE Mathematics Advanced - Question 19 - 2023 - Paper 1
Step 1
a) Sketch the graphs of the functions $f(x) = x - 1$ and $g(x) = (1 - x)(3 + x)$ showing the $x$-intercepts.
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Answer
Function f(x)=x−1:
To find the x-intercept, set f(x)=0:
x−1=0⇒x=1
This means the graph intersects the x-axis at (1,0).
Function g(x)=(1−x)(3+x):
To find the x-intercepts, set g(x)=0:
(1−x)(3+x)=0
This gives us two cases to solve:
1−x=0⇒x=1
3+x=0⇒x=−3
The graph intersects the x-axis at (−3,0) and (1,0).
Sketching the Graph:
Plot the points of intersection: (−3,0) and (1,0).
The graph of f(x) is a straight line with a slope of 1 passing through (1,0).
The graph of g(x) is a parabola opening downwards that passes through (−3,0) and (1,0).
Mark these x-intercepts clearly on the sketch.
Step 2
b) Solve the inequality $x - 1 < (1 - x)(3 + x)$.
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Answer
Set up the inequality:x−1<(1−x)(3+x)
Expand the right side:
Distributing the terms:
x−1<3+x−3x−x2
Which simplifies to:
x−1<−x2−2x+3
Rearranging the inequality:0<−x2−3x+4x2+3x−4<0
Factoring the quadratic:
The roots are found by solving:
x2+4x−1=0
Using the quadratic formula:
x=2a−b±b2−4ac=2−3±9+16=2−3±5
Therefore, the roots are:
x=1,x=−4
Test intervals:
The inequality x2+3x−4<0 holds between the roots:
−4<x<1
