Given functions: f(x) = 2x + 3 where x [−1, 2) and g(x) = x2. Determine g(sqrt(3)), f⁻¹(x), a if g(a+2)…

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Given functions: f(x) = 2x + 3 where x [−1, 2) and g(x) = x2. Determine g(sqrt(3)), f⁻¹(x), a if g(a+2) − f(a+2) = 0, and the domain of f⁻¹(x).

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Given functions:
$f(x) = 2x + 3$ where $x \in [-1, 2)$
$g(x) = x^2$ with domain $x \in \mathbb{R}$

2.3.1 Determine the value of $g(\sqrt{3})$ .

Substitute $\sqrt{3}$ into the function $g(x)$ .
$g(x) = x^2$ $g(\sqrt{3}) = (\sqrt{3})^2$
Simplify the expression.
$g(\sqrt{3}) = 3$
The value of $g(\sqrt{3})$ is $\boxed{3}$ .

2.3.2 Determine the equation of $f^{-1}(x)$ in the form $y=...$ .
Replace $f(x)$ with $y$ .
$y = 2x + 3$
Swap $x$ and $y$ to find the inverse.
$x = 2y + 3$
Solve for $y$ .
$x - 3 = 2y$ $y = \frac{x - 3}{2}$
The equation of $f^{-1}(x)$ is $\boxed{y = \frac{x - 3}{2}}$ .

2.3.3 Determine the value of $a$ if $g(a+2) - f(a+2) = 0$ .
Substitute $(a+2)$ into $g(x)$ and $f(x)$ .
$g(a+2) = (a+2)^2$ $f(a+2) = 2(a+2) + 3$
Set up the equation $g(a+2) - f(a+2) = 0$ .
$(a+2)^2 - (2(a+2) + 3) = 0$
Expand and simplify the equation.
$(a^2 + 4a + 4) - (2a + 4 + 3) = 0$ $a^2 + 4a + 4 - 2a - 7 = 0$ $a^2 + 2a - 3 = 0$
Factor the quadratic equation.
$(a+3)(a-1) = 0$
Solve for $a$ .
$a+3 = 0 \implies a = -3$ $a-1 = 0 \implies a = 1$
The values of $a$ are $\boxed{-3 \text{ or } 1}$ .

2.3.4 Write down the domain for $f^{-1}(x)$ .
Determine the range of $f(x)$ .
The domain of $f^{-1}(x)$ is the range of $f(x)$ .
The domain of $f(x)$ is $x \in [-1, 2)$ .
For $f(x) = 2x+3$ :
When $x = -1$ , $f(-1) = 2(-1) + 3 = -2 + 3 = 1$ .
When $x = 2$ , $f(2) = 2(2) + 3 = 4 + 3 = 7$ .
Since $f(x)$ is a linear function with a positive slope, its range is $[1, 7)$ .
State the domain of $f^{-1}(x)$ .
The domain for $f^{-1}(x)$ is $\boxed{x \in [1, 7)}$ .

2.3.5 Prove that $f(f^{-1}(x)) = f^{-1}(f(x))$ .
Calculate $f(f^{-1}(x))$ .
We have $f(x) = 2x+3$ and $f^{-1}(x) = \frac{x-3}{2}$ .
$f(f^{-1}(x)) = f\left(\frac{x-3}{2}\right)$ $f(f^{-1}(x)) = 2\left(\frac{x-3}{2}\right) + 3$ $f(f^{-1}(x)) = (x-3) + 3$ $f(f^{-1}(x)) = x$
Calculate $f^{-1}(f(x))$ .
$f^{-1}(f(x)) = f^{-1}(2x+3)$ $f^{-1}(f(x)) = \frac{(2x+3)-3}{2}$ $f^{-1}(f(x)) = \frac{2x}{2}$ $f^{-1}(f(x)) = x$
Compare the results.
Since $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ , it is proven that $f(f^{-1}(x)) = f^{-1}(f(x))$ .

2.3.6 Draw a neat graph of $f(x)$ and $f^{-1}(x)$ on the same system of axes on ADDENDUM A (attached), clearly showing the nature of the start and endpoints on the graphs. Also show the line about which $f(x)$ reflects $f^{-1}(x)$ and label it.

To draw the graph:
- Graph of $f(x) = 2x+3$ :
  
  Plot the starting point: For $x=-1$ , $f(-1) = 2(-1)+3 = 1$ . This is a closed circle at $(-1, 1)$ because $x \ge -1$ .
  
  Plot the ending point: For $x=2$ , $f(2) = 2(2)+3 = 7$ . This is an open circle at $(2, 7)$ because $x < 2$ .
  
  Draw a straight line connecting these two points.
- Graph of $f^{-1}(x) = \frac{x-3}{2}$ :
  
  The domain of $f^{-1}(x)$ is $x \in [1, 7)$ .
  
  Plot the starting point: For $x=1$ , $f^{-1}(1) = \frac{1-3}{2} = -1$ . This is a closed circle at $(1, -1)$ .
  
  Plot the ending point: For $x=7$ , $f^{-1}(7) = \frac{7-3}{2} = 2$ . This is an open circle at $(7, 2)$ .
  
  Draw a straight line connecting these two points.
- Line of reflection:
  
  Draw the line $y=x$ . This line acts as the axis of symmetry between $f(x)$ and $f^{-1}(x)$ . Label this line as $y=x$ .

Given functions: f(x) = 2x + 3 where x [-1, 2) and g(x) = x2. Determine g(sqrt(3)), f⁻¹(x), a if g(a+2) − f(a+2) = 0, and the domain of f⁻¹(x).

Step-by-step solution

2.3.1 Determine the value of $g(\sqrt{3})$ .

Substitute $\sqrt{3}$ into the function $g(x)$ .

Simplify the expression.

2.3.2 Determine the equation of $f^{-1}(x)$ in the form $y=...$ .

Replace $f(x)$ with $y$ .

Swap $x$ and $y$ to find the inverse.

Solve for $y$ .

2.3.3 Determine the value of $a$ if $g(a+2) - f(a+2) = 0$ .

Substitute $(a+2)$ into $g(x)$ and $f(x)$ .

Set up the equation $g(a+2) - f(a+2) = 0$ .

Expand and simplify the equation.

Factor the quadratic equation.

Solve for $a$ .

2.3.4 Write down the domain for $f^{-1}(x)$ .

Determine the range of $f(x)$ .

State the domain of $f^{-1}(x)$ .

2.3.5 Prove that $f(f^{-1}(x)) = f^{-1}(f(x))$ .

Calculate $f(f^{-1}(x))$ .

Calculate $f^{-1}(f(x))$ .

Compare the results.

2.3.6 Draw a neat graph of $f(x)$ and $f^{-1}(x)$ on the same system of axes on ADDENDUM A (attached), clearly showing the nature of the start and endpoints on the graphs. Also show the line about which $f(x)$ reflects $f^{-1}(x)$ and label it.

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More Mathematics Questions

Given functions: f(x) = 2x + 3 where x [-1, 2) and g(x) = x2. Determine g(sqrt(3)), f⁻¹(x), a if g(a+2) − f(a+2) = 0, and the domain of f⁻¹(x).

Step-by-step solution

2.3.1 Determine the value of g(3)g(\sqrt{3})g(3​).

Substitute 3\sqrt{3}3​ into the function g(x)g(x)g(x).

Simplify the expression.

2.3.2 Determine the equation of f−1(x)f^{-1}(x)f−1(x) in the form y=...y=...y=....

Replace f(x)f(x)f(x) with yyy.

Swap xxx and yyy to find the inverse.

Solve for yyy.

2.3.3 Determine the value of aaa if g(a+2)−f(a+2)=0g(a+2) - f(a+2) = 0g(a+2)−f(a+2)=0.

Substitute (a+2)(a+2)(a+2) into g(x)g(x)g(x) and f(x)f(x)f(x).

Set up the equation g(a+2)−f(a+2)=0g(a+2) - f(a+2) = 0g(a+2)−f(a+2)=0.

Expand and simplify the equation.

Factor the quadratic equation.

Solve for aaa.

2.3.4 Write down the domain for f−1(x)f^{-1}(x)f−1(x).

Determine the range of f(x)f(x)f(x).

State the domain of f−1(x)f^{-1}(x)f−1(x).

2.3.5 Prove that f(f−1(x))=f−1(f(x))f(f^{-1}(x)) = f^{-1}(f(x))f(f−1(x))=f−1(f(x)).

Calculate f(f−1(x))f(f^{-1}(x))f(f−1(x)).

Calculate f−1(f(x))f^{-1}(f(x))f−1(f(x)).

Compare the results.

2.3.6 Draw a neat graph of f(x)f(x)f(x) and f−1(x)f^{-1}(x)f−1(x) on the same system of axes on ADDENDUM A (attached), clearly showing the nature of the start and endpoints on the graphs. Also show the line about which f(x)f(x)f(x) reflects f−1(x)f^{-1}(x)f−1(x) and label it.

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More Mathematics Questions

2.3.1 Determine the value of $g(\sqrt{3})$ .

Substitute $\sqrt{3}$ into the function $g(x)$ .

2.3.2 Determine the equation of $f^{-1}(x)$ in the form $y=...$ .

Replace $f(x)$ with $y$ .

Swap $x$ and $y$ to find the inverse.

Solve for $y$ .

2.3.3 Determine the value of $a$ if $g(a+2) - f(a+2) = 0$ .

Substitute $(a+2)$ into $g(x)$ and $f(x)$ .

Set up the equation $g(a+2) - f(a+2) = 0$ .

Solve for $a$ .

2.3.4 Write down the domain for $f^{-1}(x)$ .

Determine the range of $f(x)$ .

State the domain of $f^{-1}(x)$ .

2.3.5 Prove that $f(f^{-1}(x)) = f^{-1}(f(x))$ .

Calculate $f(f^{-1}(x))$ .

Calculate $f^{-1}(f(x))$ .

2.3.6 Draw a neat graph of $f(x)$ and $f^{-1}(x)$ on the same system of axes on ADDENDUM A (attached), clearly showing the nature of the start and endpoints on the graphs. Also show the line about which $f(x)$ reflects $f^{-1}(x)$ and label it.