Answer :
To determine whether the function [tex]\( f(x) = 4x - 7 \)[/tex] is one-to-one, we need to examine its properties:
1. Is [tex]\( f(x) \)[/tex] one-to-one?
A function is one-to-one if it never takes the same value twice; that is, [tex]\( f(a) = f(b) \)[/tex] implies [tex]\( a = b \)[/tex]. Another way to determine if a function is one-to-one is to check its derivative. If the derivative is always positive or always negative, the function is strictly monotonic and hence one-to-one.
The derivative of [tex]\( f(x) = 4x - 7 \)[/tex] is [tex]\( f'(x) = 4 \)[/tex]. Since this derivative is a positive constant, [tex]\( f(x) \)[/tex] is always increasing. Therefore, [tex]\( f(x) \)[/tex] is one-to-one.
2. (a) Write an equation for the inverse function in the form [tex]\( y = f^{-1}(x) \)[/tex]:
To find the inverse function, we need to solve the equation [tex]\( y = 4x - 7 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ \begin{align*} y + 7 &= 4x \\ x &= \frac{y + 7}{4} \end{align*} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x + 7}{4} \][/tex]
3. (b) Graphing [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex]:
While graphing here isn't practical, you can visualize that both [tex]\( f(x) = 4x - 7 \)[/tex] and [tex]\( f^{-1}(x) = \frac{x + 7}{4} \)[/tex] are lines. Specifically:
- [tex]\( f(x) \)[/tex] is a line with a slope of 4 and a y-intercept at -7.
- [tex]\( f^{-1}(x) \)[/tex] is a line with a slope of [tex]\( \frac{1}{4} \)[/tex] and a y-intercept at [tex]\( \frac{7}{4} \)[/tex].
On the coordinate plane, these lines are mirror images of each other over the line [tex]\( y = x \)[/tex].
4. (c) Domain and Range of [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex]:
- For the function [tex]\( f(x) = 4x - 7 \)[/tex]:
- Domain: All real numbers ([tex]\(\mathbb{R}\)[/tex])
- Range: All real numbers ([tex]\(\mathbb{R}\)[/tex])
- For the inverse function [tex]\( f^{-1}(x) = \frac{x + 7}{4} \)[/tex]:
- Domain: All real numbers ([tex]\(\mathbb{R}\)[/tex])
- Range: All real numbers ([tex]\(\mathbb{R}\)[/tex])
Since both functions are linear and their slopes are non-zero, their domains and ranges cover all real numbers.
Summary:
- The function [tex]\( f(x) = 4x - 7 \)[/tex] is one-to-one.
- The inverse function is [tex]\( f^{-1}(x) = \frac{x + 7}{4} \)[/tex].
- The domain and range of both [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex] are all real numbers ([tex]\(\mathbb{R}\)[/tex]).
1. Is [tex]\( f(x) \)[/tex] one-to-one?
A function is one-to-one if it never takes the same value twice; that is, [tex]\( f(a) = f(b) \)[/tex] implies [tex]\( a = b \)[/tex]. Another way to determine if a function is one-to-one is to check its derivative. If the derivative is always positive or always negative, the function is strictly monotonic and hence one-to-one.
The derivative of [tex]\( f(x) = 4x - 7 \)[/tex] is [tex]\( f'(x) = 4 \)[/tex]. Since this derivative is a positive constant, [tex]\( f(x) \)[/tex] is always increasing. Therefore, [tex]\( f(x) \)[/tex] is one-to-one.
2. (a) Write an equation for the inverse function in the form [tex]\( y = f^{-1}(x) \)[/tex]:
To find the inverse function, we need to solve the equation [tex]\( y = 4x - 7 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ \begin{align*} y + 7 &= 4x \\ x &= \frac{y + 7}{4} \end{align*} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x + 7}{4} \][/tex]
3. (b) Graphing [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex]:
While graphing here isn't practical, you can visualize that both [tex]\( f(x) = 4x - 7 \)[/tex] and [tex]\( f^{-1}(x) = \frac{x + 7}{4} \)[/tex] are lines. Specifically:
- [tex]\( f(x) \)[/tex] is a line with a slope of 4 and a y-intercept at -7.
- [tex]\( f^{-1}(x) \)[/tex] is a line with a slope of [tex]\( \frac{1}{4} \)[/tex] and a y-intercept at [tex]\( \frac{7}{4} \)[/tex].
On the coordinate plane, these lines are mirror images of each other over the line [tex]\( y = x \)[/tex].
4. (c) Domain and Range of [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex]:
- For the function [tex]\( f(x) = 4x - 7 \)[/tex]:
- Domain: All real numbers ([tex]\(\mathbb{R}\)[/tex])
- Range: All real numbers ([tex]\(\mathbb{R}\)[/tex])
- For the inverse function [tex]\( f^{-1}(x) = \frac{x + 7}{4} \)[/tex]:
- Domain: All real numbers ([tex]\(\mathbb{R}\)[/tex])
- Range: All real numbers ([tex]\(\mathbb{R}\)[/tex])
Since both functions are linear and their slopes are non-zero, their domains and ranges cover all real numbers.
Summary:
- The function [tex]\( f(x) = 4x - 7 \)[/tex] is one-to-one.
- The inverse function is [tex]\( f^{-1}(x) = \frac{x + 7}{4} \)[/tex].
- The domain and range of both [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex] are all real numbers ([tex]\(\mathbb{R}\)[/tex]).
