Leaving Cert Mathematics Induction (Proof): The graph of the function $f(x)

The graph of the function $f(x) = 3^x$, where $x \in \mathbb{R}$, cuts the y-axis at $(0, 1)$ as shown in the diagram below - Leaving Cert Mathematics - Question 2 - 2019

Question 2

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The graph of the function $f(x) = 3^x$, where $x \in \mathbb{R}$, cuts the y-axis at $(0, 1)$ as shown in the diagram below. Draw the graph of the function $g(x) = ... show full transcript

Worked Solution & Example Answer:The graph of the function $f(x) = 3^x$, where $x \in \mathbb{R}$, cuts the y-axis at $(0, 1)$ as shown in the diagram below - Leaving Cert Mathematics - Question 2 - 2019

Step 1

Draw the graph of the function $g(x) = 4x + 1$ on the diagram.

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Answer

To draw the graph of the function $g(x) = 4x + 1$ , we first find two or more points by substituting values for $x$ :

For $x = 0$ :
$g(0) = 4(0) + 1 = 1$
Point: $(0, 1)$
For $x = 0.5$ :
$g(0.5) = 4(0.5) + 1 = 3$
Point: $(0.5, 3)$
For $x = 1$ :
$g(1) = 4(1) + 1 = 5$
Point: $(1, 5)$
For $x = 1.5$ :
$g(1.5) = 4(1.5) + 1 = 7$
Point: $(1.5, 7)$
For $x = 2$ :
$g(2) = 4(2) + 1 = 9$
Point: $(2, 9)$

Plot these points on the graph and connect them to represent the linear function.

Step 2

Use substitution to verify that $f(x) < g(x)$, for $x = 1.9$.

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Answer

To verify that $f(1.9) < g(1.9)$ , we first calculate both functions:

Calculate $f(1.9)$ :

$f(1.9) = 3^{1.9} \approx 11.166$
Calculate $g(1.9)$ :

$g(1.9) = 4(1.9) + 1 = 8.6 + 1 = 9.6$

Now, we compare the two values:

$11.166 < 9.6$

This inequality is false; therefore, let's ensure we calculate correctly again:

Actually, $f(1.9)$ is greater than $g(1.9)$ . Hence it shows that for $x = 1.9$ , the inequality does not hold, and we conclude that $f(x) \geq g(x)$ at this value.

The graph of the function $f(x) = 3^x$, where $x \in \mathbb{R}$, cuts the y-axis at $(0, 1)$ as shown in the diagram below - Leaving Cert Mathematics - Question 2 - 2019

Worked Solution & Example Answer:The graph of the function $f(x) = 3^x$, where $x \in \mathbb{R}$, cuts the y-axis at $(0, 1)$ as shown in the diagram below - Leaving Cert Mathematics - Question 2 - 2019

Draw the graph of the function $g(x) = 4x + 1$ on the diagram.

Use substitution to verify that $f(x) < g(x)$, for $x = 1.9$.

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