Two functions f and g are defined for x ∈ ℝ as follows: $f: x \mapsto 2^{x}$ $g: x \mapsto 9x - 3x^{2} - 1.$ (a) Complete the table below, and use it to draw the graphs of f and g for $0 \leq x \leq 3$. | x | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | |----|----|-----|---|-----|----|-----|----| | f(x) | | | | | | | | | g(x) | | | | | | | | (b) Use your graphs to estimate the value(s) of x for which $2^{x} + 3x^{2} - 9x + 1 = 0$. (c) Let k be the number such that $2^{k} = 6$. Using your graph(s), or otherwise, estimate g(k).

Leaving Cert Mathematics Graphs of Functions: Two functions f and g are define

Two functions f and g are defined for x ∈ ℝ as follows: $f: x \mapsto 2^{x}$ $g: x \mapsto 9x - 3x^{2} - 1.$ (a) Complete the table below, and use it to draw the graphs of f and g for $0 \leq x \leq 3$ - Leaving Cert Mathematics - Question 5 - 2012

Question 5

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Two functions f and g are defined for x ∈ ℝ as follows: $f: x \mapsto 2^{x}$ $g: x \mapsto 9x - 3x^{2} - 1.$ (a) Complete the table below, and use it to draw the... show full transcript

Worked Solution & Example Answer:Two functions f and g are defined for x ∈ ℝ as follows: $f: x \mapsto 2^{x}$ $g: x \mapsto 9x - 3x^{2} - 1.$ (a) Complete the table below, and use it to draw the graphs of f and g for $0 \leq x \leq 3$ - Leaving Cert Mathematics - Question 5 - 2012

Step 1

Complete the table for f(x) and g(x)

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Answer

To complete the table for the functions:

For $f(x)$ :
- At $x=0$ : $f(0) = 2^0 = 1$
- At $x=0.5$ : $f(0.5) = 2^{0.5} \approx 1.414$
- At $x=1$ : $f(1) = 2^{1} = 2$
- At $x=1.5$ : $f(1.5) = 2^{1.5} \approx 2.828$
- At $x=2$ : $f(2) = 2^{2} = 4$
- At $x=2.5$ : $f(2.5) = 2^{2.5} \approx 5.657$
- At $x=3$ : $f(3) = 2^{3} = 8$
For $g(x)$ :
- At $x=0$ : $g(0) = 9(0) - 3(0)^{2} - 1 = -1$
- At $x=0.5$ : $g(0.5) = 9(0.5) - 3(0.5)^{2} - 1 \approx 2.75$
- At $x=1$ : $g(1) = 9(1) - 3(1)^{2} - 1 = 5$
- At $x=1.5$ : $g(1.5) = 9(1.5) - 3(1.5)^{2} - 1 \approx 5.75$
- At $x=2$ : $g(2) = 9(2) - 3(2)^{2} - 1 = 7$
- At $x=2.5$ : $g(2.5) = 9(2.5) - 3(2.5)^{2} - 1 \approx 7.25$
- At $x=3$ : $g(3) = 9(3) - 3(3)^{2} - 1 = 6$ .

Step 2

Use your graphs to estimate the value(s) of x for which $2^{x} + 3x^{2} - 9x + 1 = 0$

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Answer

To estimate the values of x where the functions intersect, rearranging gives us:

$2^{x} = 9x - 3x^{2} + 1$

Plotting $2^{x}$ and $g(x)$ on the same graph, we look for points of intersection. Based on the graph, the estimated values of x at the two intersection points are approximately:

$x \approx 0.26$
$x \approx 2.15$

Step 3

Let k be the number such that $2^{k} = 6$. Using your graph(s), estimate g(k).

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Answer

To find the value of k, we need $2^{k} = 6$ . On the graph of $f(x)$ , we can draw a horizontal line at $y=6$ and find where it intersects.

Through estimation, we find that this occurs when $x \approx 2.59$ . At this x-value, we can determine $g(k)$ :

$g(k) = 9(2.59) - 3(2.59)^{2} - 1 \approx 2.19$

To confirm, we can approximate further:

To find $g(2.59)$ $g (2.59)$ :
- Calculate $g(2.59)$ for a more precise intersection:

$g(2.59) = 9(2.59) - 3(2.59)^{2} - 1 \approx 2.1857$

Two functions f and g are defined for x ∈ ℝ as follows: $f: x \mapsto 2^{x}$ $g: x \mapsto 9x - 3x^{2} - 1.$ (a) Complete the table below, and use it to draw the graphs of f and g for $0 \leq x \leq 3$ - Leaving Cert Mathematics - Question 5 - 2012

Worked Solution & Example Answer:Two functions f and g are defined for x ∈ ℝ as follows: $f: x \mapsto 2^{x}$ $g: x \mapsto 9x - 3x^{2} - 1.$ (a) Complete the table below, and use it to draw the graphs of f and g for $0 \leq x \leq 3$ - Leaving Cert Mathematics - Question 5 - 2012

Complete the table for f(x) and g(x)

Use your graphs to estimate the value(s) of x for which $2^{x} + 3x^{2} - 9x + 1 = 0$

Let k be the number such that $2^{k} = 6$. Using your graph(s), estimate g(k).

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