Sketched below are the graphs of f(x) = k/x and g(x) = 2x with coordinates of A (2; 4). What is the…

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Sketched below are the graphs of f(x) = k/x and g(x) = 2x with coordinates of A (2; 4). What is the mathematical name of the graph of g(x)?

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Here are the solutions to the remaining questions:

2.2.3 Write down the domain for $f(x)$ .
From the previous question, $f(x) = x+2$ . This is a linear function.
The domain of a linear function includes all real numbers.
The domain for $f(x)$ is $\boxed{\mathbf{x \in \mathbb{R}}}$ .

2.2.4 Is $f(x)$ continuous or discontinuous? Give a reason for your answer.
The function $f(x)$ is $\boxed{\mathbf{continuous}}$ .
Reason: The graph of $f(x)=x+2$ is a straight line with no breaks, jumps, or holes. It can be drawn without lifting the pen.

2.3 Sketched below are the graphs of $f(x) = \frac{k}{x}$ and $g(x) = 2^x$ with coordinates of A $(2, 4)$ .

2.3.1 What is the mathematical name of the graph of $g(x)$ ?
The graph of $g(x)=2^x$ is an $\boxed{\mathbf{exponential \ function}}$ .

2.3.2 Determine the equation of the y-intercept of $g(x)$ .

To find the y-intercept, set $x=0$ in the function $g(x)=2^x$ .
$g(0) = 2^0$ $g(0) = 1$
The y-intercept of $g(x)$ is $\boxed{\mathbf{(0, 1)}}$ .

2.3.3 Is $g(x)$ an increasing or decreasing function?
Observe the graph of $g(x)=2^x$ .
As the $x$ -values increase, the $y$ -values also increase.
Alternatively, for an exponential function $y=a^x$ , if the base $a > 1$ , the function is increasing. Here, $a=2$ , which is greater than 1.
The function $g(x)$ is an $\boxed{\mathbf{increasing \ function}}$ .

2.3.4 Determine the equation of $f(x)$ .
The function is given as $f(x) = \frac{k}{x}$ .
The graph shows that point A $(2, 4)$ lies on $f(x)$ .
Substitute $x=2$ and $f(x)=4$ into the equation to find $k$ .
$4 = \frac{k}{2}$
Solve for $k$ .
$k = 4 \times 2$ $k = 8$
Substitute the value of $k$ back into the function's equation.
$f(x) = \frac{8}{x}$
The equation of $f(x)$ is $\boxed{\mathbf{f(x) = \frac{8}{x}}}$ .

2.3.5 Determine the domain and the range of $f(x)$ .
For the function $f(x) = \frac{8}{x}$ , the denominator cannot be zero.
The domain is all real numbers except $x=0$ .
$\text{Domain: } \{x \in \mathbb{R} \mid x \neq 0\} \text{ or } (-\infty, 0) \cup (0, \infty)$
For the function $f(x) = \frac{8}{x}$ , the function can take any real value except zero.
The range is all real numbers except $y=0$ .
$\text{Range: } \{y \in \mathbb{R} \mid y \neq 0\} \text{ or } (-\infty, 0) \cup (0, \infty)$
The domain is $\boxed{\mathbf{x \in \mathbb{R}, x \neq 0}}$ and the range is $\boxed{\mathbf{y \in \mathbb{R}, y \neq 0}}$ .

Sketched below are the graphs of f(x) = k/x and g(x) = 2x with coordinates of A (2; 4). What is the mathematical name of the graph of g(x)?

Step-by-step solution

To find the y-intercept, set $x=0$ in the function $g(x)=2^x$ .

Observe the graph of $g(x)=2^x$ .

The function is given as $f(x) = \frac{k}{x}$ .

The graph shows that point A $(2, 4)$ lies on $f(x)$ .

Solve for $k$ .

Substitute the value of $k$ back into the function's equation.

For the function $f(x) = \frac{8}{x}$ , the denominator cannot be zero.

For the function $f(x) = \frac{8}{x}$ , the function can take any real value except zero.

Still stuck on this one?

More Mathematics Questions

Sketched below are the graphs of f(x) = k/x and g(x) = 2x with coordinates of A (2; 4). What is the mathematical name of the graph of g(x)?

Step-by-step solution

To find the y-intercept, set x=0x=0x=0 in the function g(x)=2xg(x)=2^xg(x)=2x.

Observe the graph of g(x)=2xg(x)=2^xg(x)=2x.

The function is given as f(x)=kxf(x) = \frac{k}{x}f(x)=xk​.

The graph shows that point A (2,4)(2, 4)(2,4) lies on f(x)f(x)f(x).

Solve for kkk.

Substitute the value of kkk back into the function's equation.

For the function f(x)=8xf(x) = \frac{8}{x}f(x)=x8​, the denominator cannot be zero.

For the function f(x)=8xf(x) = \frac{8}{x}f(x)=x8​, the function can take any real value except zero.

Still stuck on this one?

More Mathematics Questions

To find the y-intercept, set $x=0$ in the function $g(x)=2^x$ .

Observe the graph of $g(x)=2^x$ .

The function is given as $f(x) = \frac{k}{x}$ .

The graph shows that point A $(2, 4)$ lies on $f(x)$ .

Solve for $k$ .

Substitute the value of $k$ back into the function's equation.

For the function $f(x) = \frac{8}{x}$ , the denominator cannot be zero.

For the function $f(x) = \frac{8}{x}$ , the function can take any real value except zero.