Answer :
To address the given question, let's analyze the key features of both functions independently, and then identify what they have in common.
### Analysis of Polynomial Function [tex]\( f(x) = 2x^2 - 3x + 5 \)[/tex]:
1. Domain:
- The domain of a polynomial function is all real numbers, which can be denoted as [tex]\((-\infty, \infty)\)[/tex].
2. Range:
- The range of the function can be determined by analyzing its behavior. Since it is a quadratic function and opens upwards (leading coefficient is positive), it attains a minimum value and extends to infinity.
- The minimum value of [tex]\( f(x) \)[/tex] occurs at [tex]\( x = \frac{3}{4} \)[/tex]. The corresponding output (minimum value) is [tex]\( f\left(\frac{3}{4}\right) = \frac{31}{8} \)[/tex], which approximates to 3.875.
- Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\([3.875, \infty)\)[/tex].
3. Intercepts:
- y-intercept:
- Setting [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 5 \)[/tex]. Thus, the y-intercept is 5.
- x-intercepts:
- These can be found by solving [tex]\( 2x^2 - 3x + 5 = 0 \)[/tex]. Since the discriminant [tex]\( b^2 - 4ac \)[/tex] is negative, there are no real roots, and hence no x-intercepts.
4. Asymptotes:
- Polynomial functions do not have asymptotes.
5. Behavior at Infinity:
- As [tex]\( x \to \pm \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
### Analysis of Exponential Function [tex]\( g(x) = 2^x - 5 \)[/tex]:
1. Domain:
- The domain of an exponential function of the form [tex]\( 2^x \)[/tex] is all real numbers, [tex]\((-\infty, \infty)\)[/tex].
2. Range:
- The range of [tex]\( g(x) = 2^x - 5 \)[/tex] stems from the values [tex]\( 2^x \)[/tex] takes. [tex]\( 2^x > 0 \)[/tex] for all real [tex]\( x \)[/tex].
- Thus, [tex]\( g(x) \)[/tex] lies in the interval [tex]\( (-5, \infty) \)[/tex].
3. Intercepts:
- y-intercept:
- Setting [tex]\( x = 0 \)[/tex], [tex]\( g(0) = -4 \)[/tex]. Thus, the y-intercept is -4.
- x-intercepts:
- These can be found by solving [tex]\( 2^x - 5 = 0 \)[/tex]. This results in [tex]\( 2^x = 5 \)[/tex], leading to [tex]\( x = \log_2(5) \)[/tex].
4. Asymptotes:
- The horizontal asymptote for the function [tex]\( g(x) = 2^x - 5 \)[/tex] is [tex]\( y = -5 \)[/tex].
5. Behavior at Infinity:
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to -5 \)[/tex].
### Identifying Common Features:
1. Domain:
- Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain: all real numbers [tex]\((-\infty, \infty)\)[/tex].
This common feature is highlighted as follows:
- Domain of [tex]\( f(x) \)[/tex]: [tex]\((-\infty, \infty)\)[/tex]
- Domain of [tex]\( g(x) \)[/tex]: [tex]\((-\infty, \infty)\)[/tex]
### Conclusion:
Therefore, the key feature that the polynomial function [tex]\( f(x) = 2x^2 - 3x + 5 \)[/tex] and the exponential function [tex]\( g(x) = 2^x - 5 \)[/tex] have in common is the domain of all real numbers [tex]\((-\infty, \infty)\)[/tex].
### Analysis of Polynomial Function [tex]\( f(x) = 2x^2 - 3x + 5 \)[/tex]:
1. Domain:
- The domain of a polynomial function is all real numbers, which can be denoted as [tex]\((-\infty, \infty)\)[/tex].
2. Range:
- The range of the function can be determined by analyzing its behavior. Since it is a quadratic function and opens upwards (leading coefficient is positive), it attains a minimum value and extends to infinity.
- The minimum value of [tex]\( f(x) \)[/tex] occurs at [tex]\( x = \frac{3}{4} \)[/tex]. The corresponding output (minimum value) is [tex]\( f\left(\frac{3}{4}\right) = \frac{31}{8} \)[/tex], which approximates to 3.875.
- Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\([3.875, \infty)\)[/tex].
3. Intercepts:
- y-intercept:
- Setting [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 5 \)[/tex]. Thus, the y-intercept is 5.
- x-intercepts:
- These can be found by solving [tex]\( 2x^2 - 3x + 5 = 0 \)[/tex]. Since the discriminant [tex]\( b^2 - 4ac \)[/tex] is negative, there are no real roots, and hence no x-intercepts.
4. Asymptotes:
- Polynomial functions do not have asymptotes.
5. Behavior at Infinity:
- As [tex]\( x \to \pm \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
### Analysis of Exponential Function [tex]\( g(x) = 2^x - 5 \)[/tex]:
1. Domain:
- The domain of an exponential function of the form [tex]\( 2^x \)[/tex] is all real numbers, [tex]\((-\infty, \infty)\)[/tex].
2. Range:
- The range of [tex]\( g(x) = 2^x - 5 \)[/tex] stems from the values [tex]\( 2^x \)[/tex] takes. [tex]\( 2^x > 0 \)[/tex] for all real [tex]\( x \)[/tex].
- Thus, [tex]\( g(x) \)[/tex] lies in the interval [tex]\( (-5, \infty) \)[/tex].
3. Intercepts:
- y-intercept:
- Setting [tex]\( x = 0 \)[/tex], [tex]\( g(0) = -4 \)[/tex]. Thus, the y-intercept is -4.
- x-intercepts:
- These can be found by solving [tex]\( 2^x - 5 = 0 \)[/tex]. This results in [tex]\( 2^x = 5 \)[/tex], leading to [tex]\( x = \log_2(5) \)[/tex].
4. Asymptotes:
- The horizontal asymptote for the function [tex]\( g(x) = 2^x - 5 \)[/tex] is [tex]\( y = -5 \)[/tex].
5. Behavior at Infinity:
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to -5 \)[/tex].
### Identifying Common Features:
1. Domain:
- Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain: all real numbers [tex]\((-\infty, \infty)\)[/tex].
This common feature is highlighted as follows:
- Domain of [tex]\( f(x) \)[/tex]: [tex]\((-\infty, \infty)\)[/tex]
- Domain of [tex]\( g(x) \)[/tex]: [tex]\((-\infty, \infty)\)[/tex]
### Conclusion:
Therefore, the key feature that the polynomial function [tex]\( f(x) = 2x^2 - 3x + 5 \)[/tex] and the exponential function [tex]\( g(x) = 2^x - 5 \)[/tex] have in common is the domain of all real numbers [tex]\((-\infty, \infty)\)[/tex].
