Answer :
Let's solve the given problems step by step:
### Part (a): Domain and Range of [tex]\( R = \{(x, y) \mid x + y \leq 7\} \)[/tex]
To find the domain and range of [tex]\( R \)[/tex]:
1. Domain:
- The domain consists of all possible [tex]\( x \)[/tex] values for which there exists some [tex]\( y \)[/tex] such that [tex]\( x + y \leq 7 \)[/tex].
- If [tex]\( x = 0 \)[/tex], then [tex]\( y \)[/tex] can be any value from [tex]\(-\infty\)[/tex] to 7.
- If [tex]\( y = 0 \)[/tex], then [tex]\( x \)[/tex] can be any value from [tex]\(-\infty\)[/tex] to 7.
- Therefore, the domain includes all [tex]\( x \)[/tex] values from [tex]\(-\infty\)[/tex] to 7 (because [tex]\( x \)[/tex] can also continue to [tex]\(-\infty\)[/tex] for satisfying the condition with corresponding positive [tex]\( y \)[/tex] values).
Hence, the domain of [tex]\( R \)[/tex] is [tex]\((- \infty, 7]\)[/tex].
2. Range:
- The range consists of all possible [tex]\( y \)[/tex] values for which there exists some [tex]\( x \)[/tex] such that [tex]\( x + y \leq 7 \)[/tex].
- From the above analysis (domain), if [tex]\( y = 0 \)[/tex], [tex]\( x \)[/tex] can be from [tex]\(-\infty\)[/tex] to 7.
- Thus, corresponding to domain values, [tex]\( y \)[/tex] can go from [tex]\(-\infty\)[/tex] to 7.
Hence, the range of [tex]\( R \)[/tex] is [tex]\((- \infty, 7]\)[/tex].
3. Function Check:
- A relation [tex]\( R \)[/tex] is a function if for each [tex]\( x \)[/tex] there is exactly one corresponding [tex]\( y \)[/tex].
- In this case, for a given [tex]\( x \)[/tex], [tex]\( y \)[/tex] can take multiple values as long as [tex]\( x + y \leq 7 \)[/tex].
Therefore, [tex]\( R \)[/tex] is _not_ a function.
### Part (b): Pre-image for [tex]\( f(x) = x^2 - 3 \)[/tex] when one image is 22
Given [tex]\( f(x) = x^2 - 3 \)[/tex] and [tex]\( f(x) = 22 \)[/tex]:
1. Set the equation: [tex]\( x^2 - 3 = 22 \)[/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 3 = 22 \\ x^2 = 25 \\ x = \pm 5 \][/tex]
Therefore, the pre-images of [tex]\( 22 \)[/tex] are [tex]\( x = 5 \)[/tex] and [tex]\( x = -5 \)[/tex].
### Part (c): Pre-image for [tex]\( g(x) = x^2 - 2x + 1 \)[/tex] when one image is 1
Given [tex]\( g(x) = x^2 - 2x + 1 \)[/tex] and [tex]\( g(x) = 1 \)[/tex]:
1. Set the equation: [tex]\( x^2 - 2x + 1 = 1 \)[/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 2x + 1 - 1 = 0 \\ x^2 - 2x = 0 \\ x(x - 2) = 0 \\ x = 0 \ \text{or} \ x = 2 \][/tex]
Therefore, the pre-images of [tex]\( 1 \)[/tex] are [tex]\( x = 0 \)[/tex] and [tex]\( x = 2 \)[/tex].
### Part (d): Use of Functions in Daily Life
Discussion/Report on Uses of Functions in Daily Life:
Functions are essential mathematical tools used in various real-life scenarios:
1. Finance: Interest calculations, amortization schedules, and other financial modeling use functions to relate interest rates, time periods, and payments.
2. Engineering: Signal processing, control systems, and performance analysis leverage functions to model and predict system behaviors.
3. Medicine: Dosage calculations and growth charts use functions to relate patient data with appropriate treatments or developmental metrics.
4. Economics: Supply and demand curves, cost functions, and utility functions help understand market dynamics and consumer behavior.
5. Technology: Algorithms and computer programs are fundamentally built upon functions that process inputs to yield outputs.
6. Daily Estimations: Travel time estimation depending on speed, budgeting expenses based on income, etc.
In conclusion, understanding and applying functions help solve practical problems and improve decision-making in everyday life. This collaborative exploration sheds light on the far-reaching implications of mathematical functions across different fields and personal activities.
### Part (a): Domain and Range of [tex]\( R = \{(x, y) \mid x + y \leq 7\} \)[/tex]
To find the domain and range of [tex]\( R \)[/tex]:
1. Domain:
- The domain consists of all possible [tex]\( x \)[/tex] values for which there exists some [tex]\( y \)[/tex] such that [tex]\( x + y \leq 7 \)[/tex].
- If [tex]\( x = 0 \)[/tex], then [tex]\( y \)[/tex] can be any value from [tex]\(-\infty\)[/tex] to 7.
- If [tex]\( y = 0 \)[/tex], then [tex]\( x \)[/tex] can be any value from [tex]\(-\infty\)[/tex] to 7.
- Therefore, the domain includes all [tex]\( x \)[/tex] values from [tex]\(-\infty\)[/tex] to 7 (because [tex]\( x \)[/tex] can also continue to [tex]\(-\infty\)[/tex] for satisfying the condition with corresponding positive [tex]\( y \)[/tex] values).
Hence, the domain of [tex]\( R \)[/tex] is [tex]\((- \infty, 7]\)[/tex].
2. Range:
- The range consists of all possible [tex]\( y \)[/tex] values for which there exists some [tex]\( x \)[/tex] such that [tex]\( x + y \leq 7 \)[/tex].
- From the above analysis (domain), if [tex]\( y = 0 \)[/tex], [tex]\( x \)[/tex] can be from [tex]\(-\infty\)[/tex] to 7.
- Thus, corresponding to domain values, [tex]\( y \)[/tex] can go from [tex]\(-\infty\)[/tex] to 7.
Hence, the range of [tex]\( R \)[/tex] is [tex]\((- \infty, 7]\)[/tex].
3. Function Check:
- A relation [tex]\( R \)[/tex] is a function if for each [tex]\( x \)[/tex] there is exactly one corresponding [tex]\( y \)[/tex].
- In this case, for a given [tex]\( x \)[/tex], [tex]\( y \)[/tex] can take multiple values as long as [tex]\( x + y \leq 7 \)[/tex].
Therefore, [tex]\( R \)[/tex] is _not_ a function.
### Part (b): Pre-image for [tex]\( f(x) = x^2 - 3 \)[/tex] when one image is 22
Given [tex]\( f(x) = x^2 - 3 \)[/tex] and [tex]\( f(x) = 22 \)[/tex]:
1. Set the equation: [tex]\( x^2 - 3 = 22 \)[/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 3 = 22 \\ x^2 = 25 \\ x = \pm 5 \][/tex]
Therefore, the pre-images of [tex]\( 22 \)[/tex] are [tex]\( x = 5 \)[/tex] and [tex]\( x = -5 \)[/tex].
### Part (c): Pre-image for [tex]\( g(x) = x^2 - 2x + 1 \)[/tex] when one image is 1
Given [tex]\( g(x) = x^2 - 2x + 1 \)[/tex] and [tex]\( g(x) = 1 \)[/tex]:
1. Set the equation: [tex]\( x^2 - 2x + 1 = 1 \)[/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 2x + 1 - 1 = 0 \\ x^2 - 2x = 0 \\ x(x - 2) = 0 \\ x = 0 \ \text{or} \ x = 2 \][/tex]
Therefore, the pre-images of [tex]\( 1 \)[/tex] are [tex]\( x = 0 \)[/tex] and [tex]\( x = 2 \)[/tex].
### Part (d): Use of Functions in Daily Life
Discussion/Report on Uses of Functions in Daily Life:
Functions are essential mathematical tools used in various real-life scenarios:
1. Finance: Interest calculations, amortization schedules, and other financial modeling use functions to relate interest rates, time periods, and payments.
2. Engineering: Signal processing, control systems, and performance analysis leverage functions to model and predict system behaviors.
3. Medicine: Dosage calculations and growth charts use functions to relate patient data with appropriate treatments or developmental metrics.
4. Economics: Supply and demand curves, cost functions, and utility functions help understand market dynamics and consumer behavior.
5. Technology: Algorithms and computer programs are fundamentally built upon functions that process inputs to yield outputs.
6. Daily Estimations: Travel time estimation depending on speed, budgeting expenses based on income, etc.
In conclusion, understanding and applying functions help solve practical problems and improve decision-making in everyday life. This collaborative exploration sheds light on the far-reaching implications of mathematical functions across different fields and personal activities.
