SSCE HSC Mathematics Advanced Absolute value functions: Sketch the graphs of the functio

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. (b) Hence, or otherwise, solve the inequality $x - 1 < (1 ... 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

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

To sketch the graphs, we find the $x$ -intercepts of both functions:

For $f(x)$ :
- Set $f(x) = 0$ : $x - 1 = 0$ $x = 1$
- The graph is a straight line; it intersects the $x$ -axis at $(1, 0)$ .
For $g(x)$ :
- Set $g(x) = 0$ : $(1 - x)(3 + x) = 0$
- This gives two values:
  - $1 - x = 0 ightarrow x = 1$
  - $3 + x = 0 ightarrow x = -3$
- The graph is a downward-opening parabola; it intersects the $x$ -axis at $(-3, 0)$ and $(1, 0)$ .
Sketch the graphs:
- Plot the points $(1, 0)$ and $(-3, 0)$ on the graph.
- Draw the line $f(x)$ from $(- ext{infinity}, -1)$ to $(1, 0)$ and beyond.
- Draw the parabola $g(x)$ between $(-3, 0)$ and $(1, 0)$ .

The sketch should clearly show the features of both curves.

Step 2

Hence, or otherwise, solve the inequality $x - 1 < (1 - x)(3 + x)$.

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Answer

To solve the inequality:

Expand the right-hand side:
- Start with the equation: $x - 1 < (1 - x)(3 + x)$
- Expanding gives: $x - 1 < 3 + x - 3x - x^2$ $x - 1 < -x^2 - 2x + 3$
Rearranging:
- Bringing all terms to one side yields: $x + x^2 + 2x - 3 - 1 < 0$ $x^2 + 3x - 4 < 0$
Factor the quadratic:
- We can factor this as: $(x + 4)(x - 1) < 0$
Determine the intervals:
- The roots are $x = -4$ and $x = 1$ .
- Test the intervals:
  - For $x < -4$ , values are positive.
  - For $-4 < x < 1$ , values are negative (solution interval).
  - For $x > 1$ , values are positive.

Thus, the solution to the inequality is: $-4 < x < 1$

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

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

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 $x - 1 < (1 - x)(3 + x)$.

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