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### 3. State the values of [tex]\(x\)[/tex] which must be excluded from the domains of the following functions:
(a) [tex]\(f(x) = \frac{2}{x - 2}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero:
[tex]\[ x - 2 = 0 \][/tex]
[tex]\[ x = 2 \][/tex]
So, [tex]\(x = 2\)[/tex] must be excluded.
(b) [tex]\(g(x) = \frac{x - 2}{2x - 3}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero:
[tex]\[ 2x - 3 = 0 \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
So, [tex]\(x = \frac{3}{2}\)[/tex] must be excluded.
(c) [tex]\(h(x) = \frac{5}{x^2 - x - 2}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero. Factorize the denominator:
[tex]\[ x^2 - x - 2 = (x - 2)(x + 1) \][/tex]
Set each factor to zero:
[tex]\[ x - 2 = 0 \text{ or } x + 1 = 0 \][/tex]
[tex]\[ x = 2 \text{ or } x = -1 \][/tex]
So, [tex]\(x = 2\)[/tex] and [tex]\(x = -1\)[/tex] must be excluded.
(d) [tex]\(F(x) = 3 - \frac{2}{x + 3}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero:
[tex]\[ x + 3 = 0 \][/tex]
[tex]\[ x = -3 \][/tex]
So, [tex]\(x = -3\)[/tex] must be excluded.
### 4. [tex]\(f\)[/tex] is the function 'square [tex]\(r\)[/tex] and add 2'.
(a) Write [tex]\(f\)[/tex] in the form [tex]\(f(x) = \ldots\)[/tex]
The function [tex]\(f\)[/tex] is defined as 'square [tex]\(x\)[/tex] and add 2':
[tex]\[ f(x) = x^2 + 2 \][/tex]
(b) Find [tex]\(f(1)\)[/tex], [tex]\(f(-1)\)[/tex], [tex]\(f(0)\)[/tex].
Evaluating the function at the given points:
[tex]\[ f(1) = 1^2 + 2 = 3 \][/tex]
[tex]\[ f(-1) = (-1)^2 + 2 = 3 \][/tex]
[tex]\[ f(0) = 0^2 + 2 = 2 \][/tex]
So, [tex]\( f(1) = 3 \)[/tex], [tex]\( f(-1) = 3 \)[/tex], and [tex]\( f(0) = 2 \)[/tex].
(c) If [tex]\(f(x) = 27\)[/tex], find the values of [tex]\(x\)[/tex].
Set [tex]\(f(x) = x^2 + 2 = 27\)[/tex]:
[tex]\[ x^2 + 2 = 27 \][/tex]
[tex]\[ x^2 = 25 \][/tex]
[tex]\[ x = \pm 5 \][/tex]
Thus, the values of [tex]\(x\)[/tex] are [tex]\(x = 5\)[/tex] and [tex]\(x = -5\)[/tex].
### 5. [tex]\(F\)[/tex] is the function 'add 2 to [tex]\(x\)[/tex] and then square'.
(a) Write [tex]\(F\)[/tex] in the form [tex]\(F(x) = \ldots\)[/tex]
The function [tex]\(F\)[/tex] is defined as 'add 2 to [tex]\(x\)[/tex] and then square':
[tex]\[ F(x) = (x + 2)^2 \][/tex]
(b) Find [tex]\(F(1)\)[/tex], [tex]\(F(-1)\)[/tex], [tex]\(F(0)\)[/tex].
Evaluating the function at the given points:
[tex]\[ F(1) = (1 + 2)^2 = 9 \][/tex]
[tex]\[ F(-1) = (-1 + 2)^2 = 1 \][/tex]
[tex]\[ F(0) = (0 + 2)^2 = 4 \][/tex]
So, [tex]\( F(1) = 9 \)[/tex], [tex]\( F(-1) = 1 \)[/tex], and [tex]\( F(0) = 4 \)[/tex].
(c) If [tex]\(F(x) = 25\)[/tex], find the values of [tex]\(x\)[/tex].
Set [tex]\(F(x) = (x + 2)^2 = 25\)[/tex]:
[tex]\[ (x + 2)^2 = 25 \][/tex]
Taking the square root of both sides:
[tex]\[ x + 2 = \pm 5 \][/tex]
Hence, solving for [tex]\(x\)[/tex]:
[tex]\[ x + 2 = 5 \implies x = 3 \][/tex]
[tex]\[ x + 2 = -5 \implies x = -7 \][/tex]
Thus, the values of [tex]\(x\)[/tex] are [tex]\(x = 3\)[/tex] and [tex]\(x = -7\)[/tex].
(d) Is it the same function as [tex]\(f\)[/tex] in Question 4?
No, it is not the same function as [tex]\(f\)[/tex] in Question 4.
While [tex]\(f(x) = x^2 + 2\)[/tex], [tex]\( F(x) = (x + 2)^2 \)[/tex]. These two functions are different.
### 6. If [tex]\(f(x) = 3x + 2\)[/tex], what is the value of [tex]\(x\)[/tex] which is mapped onto [tex]\(8\)[/tex]?
Set [tex]\(f(x) = 3x + 2 = 8\)[/tex]:
[tex]\[ 3x + 2 = 8 \][/tex]
[tex]\[ 3x = 6 \][/tex]
[tex]\[ x = 2 \][/tex]
Thus, [tex]\(x = 2\)[/tex] is mapped onto [tex]\(8\)[/tex].
### 7. A function such as [tex]\(f(x) = 5\)[/tex] is a constant function. State the values of [tex]\(f(0)\)[/tex], [tex]\(f(-1)\)[/tex], and [tex]\(f(5)\)[/tex].
For a constant function [tex]\(f(x) = 5\)[/tex]:
[tex]\[ f(0) = 5 \][/tex]
[tex]\[ f(-1) = 5 \][/tex]
[tex]\[ f(5) = 5 \][/tex]
Thus, the values are [tex]\(f(0) = 5\)[/tex], [tex]\(f(-1) = 5\)[/tex], and [tex]\(f(5) = 5\)[/tex].
(a) [tex]\(f(x) = \frac{2}{x - 2}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero:
[tex]\[ x - 2 = 0 \][/tex]
[tex]\[ x = 2 \][/tex]
So, [tex]\(x = 2\)[/tex] must be excluded.
(b) [tex]\(g(x) = \frac{x - 2}{2x - 3}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero:
[tex]\[ 2x - 3 = 0 \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
So, [tex]\(x = \frac{3}{2}\)[/tex] must be excluded.
(c) [tex]\(h(x) = \frac{5}{x^2 - x - 2}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero. Factorize the denominator:
[tex]\[ x^2 - x - 2 = (x - 2)(x + 1) \][/tex]
Set each factor to zero:
[tex]\[ x - 2 = 0 \text{ or } x + 1 = 0 \][/tex]
[tex]\[ x = 2 \text{ or } x = -1 \][/tex]
So, [tex]\(x = 2\)[/tex] and [tex]\(x = -1\)[/tex] must be excluded.
(d) [tex]\(F(x) = 3 - \frac{2}{x + 3}\)[/tex]
To find the values of [tex]\(x\)[/tex] which must be excluded, find the values that make the denominator zero:
[tex]\[ x + 3 = 0 \][/tex]
[tex]\[ x = -3 \][/tex]
So, [tex]\(x = -3\)[/tex] must be excluded.
### 4. [tex]\(f\)[/tex] is the function 'square [tex]\(r\)[/tex] and add 2'.
(a) Write [tex]\(f\)[/tex] in the form [tex]\(f(x) = \ldots\)[/tex]
The function [tex]\(f\)[/tex] is defined as 'square [tex]\(x\)[/tex] and add 2':
[tex]\[ f(x) = x^2 + 2 \][/tex]
(b) Find [tex]\(f(1)\)[/tex], [tex]\(f(-1)\)[/tex], [tex]\(f(0)\)[/tex].
Evaluating the function at the given points:
[tex]\[ f(1) = 1^2 + 2 = 3 \][/tex]
[tex]\[ f(-1) = (-1)^2 + 2 = 3 \][/tex]
[tex]\[ f(0) = 0^2 + 2 = 2 \][/tex]
So, [tex]\( f(1) = 3 \)[/tex], [tex]\( f(-1) = 3 \)[/tex], and [tex]\( f(0) = 2 \)[/tex].
(c) If [tex]\(f(x) = 27\)[/tex], find the values of [tex]\(x\)[/tex].
Set [tex]\(f(x) = x^2 + 2 = 27\)[/tex]:
[tex]\[ x^2 + 2 = 27 \][/tex]
[tex]\[ x^2 = 25 \][/tex]
[tex]\[ x = \pm 5 \][/tex]
Thus, the values of [tex]\(x\)[/tex] are [tex]\(x = 5\)[/tex] and [tex]\(x = -5\)[/tex].
### 5. [tex]\(F\)[/tex] is the function 'add 2 to [tex]\(x\)[/tex] and then square'.
(a) Write [tex]\(F\)[/tex] in the form [tex]\(F(x) = \ldots\)[/tex]
The function [tex]\(F\)[/tex] is defined as 'add 2 to [tex]\(x\)[/tex] and then square':
[tex]\[ F(x) = (x + 2)^2 \][/tex]
(b) Find [tex]\(F(1)\)[/tex], [tex]\(F(-1)\)[/tex], [tex]\(F(0)\)[/tex].
Evaluating the function at the given points:
[tex]\[ F(1) = (1 + 2)^2 = 9 \][/tex]
[tex]\[ F(-1) = (-1 + 2)^2 = 1 \][/tex]
[tex]\[ F(0) = (0 + 2)^2 = 4 \][/tex]
So, [tex]\( F(1) = 9 \)[/tex], [tex]\( F(-1) = 1 \)[/tex], and [tex]\( F(0) = 4 \)[/tex].
(c) If [tex]\(F(x) = 25\)[/tex], find the values of [tex]\(x\)[/tex].
Set [tex]\(F(x) = (x + 2)^2 = 25\)[/tex]:
[tex]\[ (x + 2)^2 = 25 \][/tex]
Taking the square root of both sides:
[tex]\[ x + 2 = \pm 5 \][/tex]
Hence, solving for [tex]\(x\)[/tex]:
[tex]\[ x + 2 = 5 \implies x = 3 \][/tex]
[tex]\[ x + 2 = -5 \implies x = -7 \][/tex]
Thus, the values of [tex]\(x\)[/tex] are [tex]\(x = 3\)[/tex] and [tex]\(x = -7\)[/tex].
(d) Is it the same function as [tex]\(f\)[/tex] in Question 4?
No, it is not the same function as [tex]\(f\)[/tex] in Question 4.
While [tex]\(f(x) = x^2 + 2\)[/tex], [tex]\( F(x) = (x + 2)^2 \)[/tex]. These two functions are different.
### 6. If [tex]\(f(x) = 3x + 2\)[/tex], what is the value of [tex]\(x\)[/tex] which is mapped onto [tex]\(8\)[/tex]?
Set [tex]\(f(x) = 3x + 2 = 8\)[/tex]:
[tex]\[ 3x + 2 = 8 \][/tex]
[tex]\[ 3x = 6 \][/tex]
[tex]\[ x = 2 \][/tex]
Thus, [tex]\(x = 2\)[/tex] is mapped onto [tex]\(8\)[/tex].
### 7. A function such as [tex]\(f(x) = 5\)[/tex] is a constant function. State the values of [tex]\(f(0)\)[/tex], [tex]\(f(-1)\)[/tex], and [tex]\(f(5)\)[/tex].
For a constant function [tex]\(f(x) = 5\)[/tex]:
[tex]\[ f(0) = 5 \][/tex]
[tex]\[ f(-1) = 5 \][/tex]
[tex]\[ f(5) = 5 \][/tex]
Thus, the values are [tex]\(f(0) = 5\)[/tex], [tex]\(f(-1) = 5\)[/tex], and [tex]\(f(5) = 5\)[/tex].
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