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Question 1: Find the Domain and Range of the Function, $$ f(x) = 2x + 3 $$

Solution:

We are given a Linear Function \( f(x) = 2x + 3 \)

Finding the Domain and Range:
Linear functions are defined for all real numbers. Therefore, the domain is \( \text{Domain} = (-\infty, \infty) \).
Since there are no restrictions on the outputs, and it can take any real value as \( x \) varies, the range is also all real numbers: \( \text{Range} = (-\infty, \infty) \).

 Question 2: Find the Domain and Range of the Function, $$ f(x) = -5x + 7 $$

Solution:

We are given a Linear Function \( f(x) = -5x + 7 \)

Finding the Domain and Range:
The domain of this linear function is all real numbers: \( \text{Domain} = (-\infty, \infty) \).
The range is also all real numbers, as the output can be any value depending on the input \( x \): \( \text{Range} = (-\infty, \infty) \).

 Question 3: Find the Domain and Range of the Function, $$ f(x) = x^2 - 4 $$

Solution:

We are given a Quadratic Function \( f(x) = x^2 - 4 \)

Finding the Domain:
The domain of a quadratic function is all real numbers: \( \text{Domain} = (-\infty, \infty) \).
Finding the Range:
Since the parabola opens upwards and the vertex is the lowest point, the minimum value is -4 when \( x = 0 \). Thus, the range is \( \text{Range} = [-4, \infty) \).

 Question 4: Find the Domain and Range of the Function, $$ f(x) = -(x-3)^2 + 2 $$

Solution:

We are given a Quadratic Function \( f(x) = -(x-3)^2 + 2 \)

Finding the Domain:
The domain of a quadratic function is all real numbers: \( \text{Domain} = (-\infty, \infty) \).
Finding the Range:
Since the parabola opens downwards, the vertex is the highest point at \( x = 3 \) with a maximum value of 2. Thus, the range is \( \text{Range} = (-\infty, 2] \).

 Question 5: Find the Domain and Range of the Function, $$ f(x) = x^3 - 3x $$

Solution:

We are given a Cubic Function \( f(x) = x^3 - 3x \)

Finding the Domain:
The domain of cubic functions is all real numbers: \( \text{Domain} = (-\infty, \infty) \).
Finding the Range:
Cubic functions can take any real value as \( x \) varies, so the range is also all real numbers: \( \text{Range} = (-\infty, \infty) \).

 Question 6: Find the Domain and Range of the Function, $$ f(x) = x^3 + 1 $$

Solution:

We are given a Cubic Function \( f(x) = x^3 + 1 \)

Finding the Domain:
The domain of a cubic function is all real numbers: \( \text{Domain} = (-\infty, \infty) \).
Finding the Range:
Since the function can take any value as \( x \) varies, the range is all real values: \( \text{Range} = (-\infty, \infty) \).

 Question 7: Find the Domain and Range of the Function, $$ f(x) = \log(x) $$

Solution:

We are given a Logarithmic Function \( f(x) = \log(x) \)

Finding the Domain:
Logarithmic functions are defined for positive real numbers only. Thus, the domain is \( \text{Domain} = (0, \infty) \).
Finding the Range:
The logarithmic function can take any real number as its output depending on the input value, thus \( \text{Range} = (-\infty, \infty) \).

 Question 8: Find the Domain and Range of the Function, $$ f(x) = \log(2x - 1) $$

Solution:

We are given a Logarithmic Function \( f(x) = \log(2x - 1) \)

Finding the Domain:
The function \( 2x - 1 \) must be greater than zero for the logarithm to be defined. Solving \( 2x - 1 > 0 \), we find \( x > \frac{1}{2} \). Thus, the domain is \( \text{Domain} = \left(\frac{1}{2}, \infty\right) \).
Finding the Range:
Since the logarithm of a positive number can take any real value, the range is all real numbers: \( \text{Range} = (-\infty, \infty) \).

 Question 9: Find the Domain and Range of the Function, $$ f(x) = e^x $$

Solution:

We are given an Exponential Function \( f(x) = e^x \)

Finding the Domain:
The exponential function is defined for all real numbers. Thus, the domain is \( \text{Domain} = (-\infty, \infty) \).
Finding the Range:
The exponential function never takes negative values and increases without bound as \( x \) increases. Therefore, the range is \( \text{Range} = (0, \infty) \).

 Question 10: Find the Domain and Range of the Function, $$ f(x) = 2^x - 3 $$

Solution:

We are given an Exponential Function \( f(x) = 2^x - 3 \)

Finding the Domain:
This exponential function is defined for all real numbers, so the domain is \( \text{Domain} = (-\infty, \infty) \).
Finding the Range:
As \( x \) increases, \( 2^x \) increases, and \( 2^x - 3 \) shifts the graph downward by 3 units. Thus, the range starts from -3 and goes up to infinity: \( \text{Range} = (-3, \infty) \).

 Question 11: Find the Domain and Range of the Function, $$ f(x) = \sin(x) $$

Solution:

We are given a Trigonometric Function \( f(x) = \sin(x) \)

Finding the Domain:
The sine function is defined for all real numbers, hence the domain is \( \text{Domain} = (-\infty, \infty) \).
Finding the Range:
The values of \( \sin(x) \) are bounded between -1 and 1. Therefore, the range is \( \text{Range} = [-1, 1] \).

 Question 12: Find the Domain and Range of the Function, $$ f(x) = \tan(x) $$

Solution:

We are given a Trigonometric Function \( f(x) = \tan(x) \)

Finding the Domain:
The tangent function is undefined where the cosine function is zero, which occurs at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \). Thus, the domain excludes these points: \( \text{Domain} = (-\infty, \infty) \setminus \left\{\frac{\pi}{2} + n\pi\right\} \).
Finding the Range:
The tangent function can take any real value, thus the range is \( \text{Range} = (-\infty, \infty) \).

 Question 13: Find the Domain and Range of a Function, $$ f(x) = \sinh(x) $$

Solution:

We are given the Hyperbolic Function \( \sinh x \)

1. Understanding the Hyperbolic Functions:
The definition of the hyperbolic sine function is:

  • \( \sinh x = \frac{e^x - e^{-x}}{2} \)
2. Finding the Domain:
The \( \sinh x \) function is defined for any real value of \( x \), as the exponential function is always defined.
Thus, the domain of the function is: \[ \text{Domain} = (-\infty, \infty) \] 3. Finding the Range:
Since the exponential function \( e^x \) always returns positive values, the expression \( \sinh x \) will cover all real values as \( x \) varies.
Therefore, the range of \( \sinh x \) is: \[ \text{Range of } \sinh x = (-\infty, \infty) \]

Question 14: Find the Domain and Range of a Function, $$ f(x) = \cosh(x) $$

Solution:

We are given the Hyperbolic Function \( \cosh x \)

1. Understanding the Hyperbolic Functions:
The definition of the hyperbolic cosine function is:

  • \( \cosh x = \frac{e^x + e^{-x}}{2} \)
2. Finding the Domain:
The \( \cosh x \) function is defined for any real value of \( x \), as the exponential function is always defined.
Thus, the domain of the function is: \[ \text{Domain} = (-\infty, \infty) \] 3. Finding the Range:
The hyperbolic cosine function \( \cosh x \) is always positive and symmetric about the y-axis (even function), and its minimum value is 1 at \( x = 0 \).
Therefore, the range of \( \cosh x \) is: \[ \text{Range of } \cosh x = [1, \infty) \]

Question 15: Find the Domain and Range of the Function, $$ f(x) = \frac{1}{x-2} $$

Solution:

We are given a Rational Function \( f(x) = \frac{1}{x-2} \)

Finding the Domain:
The function is undefined where the denominator is zero, i.e., \( x = 2 \). Thus, the domain excludes this point:
\[ \text{Domain} = (-\infty, 2) \cup (2, \infty) \] Finding the Range:
As the function can approach any real value except zero, the range is:
\[ \text{Range} = (-\infty, 0) \cup (0, \infty) \]

Question 16: Find the Domain and Range of the Function, $$ f(x) = \sqrt{x - 1} $$

Solution:

We are given a Radical Function \( f(x) = \sqrt{x - 1} \)

Finding the Domain:
The square root function requires the radicand to be non-negative. Thus, \( x - 1 \geq 0 \) which simplifies to \( x \geq 1 \).
\[ \text{Domain} = [1, \infty) \] Finding the Range:
The square root function produces non-negative values starting from 0. Thus, the range is:
\[ \text{Range} = [0, \infty) \]

Question 17: Find the Domain and Range of the Function, $$ f(x) = \frac{x}{x^2 - 4} $$

Solution:

We are given a Rational Function \( f(x) = \frac{x}{x^2 - 4} \)

Finding the Domain:
The denominator cannot be zero, which occurs at \( x^2 - 4 = 0 \) or \( x = \pm 2 \). Hence, the domain excludes these points:
\[= (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \] Finding the Range:
This function can take any real value as \( x \) varies, except possibly for certain limitations near the vertical asymptotes. Hence, the range is generally considered to be all real numbers, \(\mathbb{R}\).

 Question 18: Find the Domain and Range of the Function, $$ f(x) = \sqrt{9 - x^2} $$

Solution:

We are given a Radical Function \( f(x) = \sqrt{9 - x^2} \)

Finding the Domain:
The expression under the square root must be non-negative, thus \( 9 - x^2 \geq 0 \) which simplifies to \( -3 \leq x \leq 3 \).
\[ \text{Domain} = [-3, 3] \] Finding the Range:
The square root function outputs values from 0 up to the maximum value of the radicand, which is 3 at \( x = 0 \). Thus, the range is:
\[ \text{Range} = [0, 3] \]