Answer :
Certainly! Let's analyze the function [tex]\( h(x) = (x - 3)^3 + 1 \)[/tex] and assess the truth of each provided statement step-by-step to determine which are true.
### Step-by-Step Analysis:
1. Domain of the function:
The given function is in the form [tex]\( h(x) = (x - 3)^3 + 1 \)[/tex]. Cubic functions [tex]\( f(x) = x^3 \)[/tex] are defined for all real numbers without any restrictions on [tex]\( x \)[/tex]. Hence, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
[tex]\[ \text{True} \][/tex]
2. Range of the function:
Similarly, cubic functions [tex]\( f(x) = x^3 \)[/tex] can output any real number, which means the range is all real numbers. The transformation involved does not restrict output values. Hence, the range of [tex]\( h(x) \)[/tex] is also all real numbers.
[tex]\[ \text{True} \][/tex]
3. Graph translation:
The function [tex]\( h(x) \)[/tex] is derived from [tex]\( f(x) = x^3 \)[/tex] by translating it 3 units to the right (because of [tex]\( (x - 3) \)[/tex]) and 1 unit up (because of the +1). Therefore, the statement should indicate a translation to the right and up, not to the left.
[tex]\[ \text{False (should be right, not left)} \][/tex]
4. Graph crossing the [tex]\( y \)[/tex]-axis:
To find the [tex]\( y \)[/tex]-axis crossing, we set [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = (0 - 3)^3 + 1 = (-3)^3 + 1 = -27 + 1 = -26 \][/tex]
Therefore, the graph crosses the [tex]\( y \)[/tex]-axis exactly once at [tex]\( (0, -26) \)[/tex].
[tex]\[ \text{True} \][/tex]
5. Graph crossing the [tex]\( x \)[/tex]-axis:
To determine where the graph crosses the [tex]\( x \)[/tex]-axis, we set [tex]\( h(x) = 0 \)[/tex]:
[tex]\[ (x - 3)^3 + 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ (x - 3)^3 = -1 \implies x - 3 = -1 \implies x = 2 \][/tex]
The graph crosses the [tex]\( x \)[/tex]-axis exactly once at [tex]\( (2, 0) \)[/tex].
[tex]\[ \text{True (crosses once, not three times)} \][/tex]
6. Graph increasing from negative infinity to positive infinity:
Since [tex]\( f(x) = x^3 \)[/tex] is an increasing function, and the transformations do not change its overall behavior, [tex]\( h(x) = (x-3)^3 + 1 \)[/tex] will also increase from negative infinity to positive infinity.
[tex]\[ \text{True} \][/tex]
### Summary of True Statements:
- The domain of the function is all real numbers.
- The range of the function is all real numbers.
- The graph crosses the [tex]\( y \)[/tex]-axis once.
- The graph crosses the [tex]\( x \)[/tex]-axis once.
- The graph increases from negative infinity to positive infinity.
Thus, the true statements are:
- The domain of the function is all real numbers.
- The range of the function is all real numbers.
- The graph crosses the [tex]\( y \)[/tex]-axis once.
- The graph crosses the [tex]\( x \)[/tex]-axis once.
- The graph increases from negative infinity to positive infinity.
### Step-by-Step Analysis:
1. Domain of the function:
The given function is in the form [tex]\( h(x) = (x - 3)^3 + 1 \)[/tex]. Cubic functions [tex]\( f(x) = x^3 \)[/tex] are defined for all real numbers without any restrictions on [tex]\( x \)[/tex]. Hence, the domain of [tex]\( h(x) \)[/tex] is all real numbers.
[tex]\[ \text{True} \][/tex]
2. Range of the function:
Similarly, cubic functions [tex]\( f(x) = x^3 \)[/tex] can output any real number, which means the range is all real numbers. The transformation involved does not restrict output values. Hence, the range of [tex]\( h(x) \)[/tex] is also all real numbers.
[tex]\[ \text{True} \][/tex]
3. Graph translation:
The function [tex]\( h(x) \)[/tex] is derived from [tex]\( f(x) = x^3 \)[/tex] by translating it 3 units to the right (because of [tex]\( (x - 3) \)[/tex]) and 1 unit up (because of the +1). Therefore, the statement should indicate a translation to the right and up, not to the left.
[tex]\[ \text{False (should be right, not left)} \][/tex]
4. Graph crossing the [tex]\( y \)[/tex]-axis:
To find the [tex]\( y \)[/tex]-axis crossing, we set [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = (0 - 3)^3 + 1 = (-3)^3 + 1 = -27 + 1 = -26 \][/tex]
Therefore, the graph crosses the [tex]\( y \)[/tex]-axis exactly once at [tex]\( (0, -26) \)[/tex].
[tex]\[ \text{True} \][/tex]
5. Graph crossing the [tex]\( x \)[/tex]-axis:
To determine where the graph crosses the [tex]\( x \)[/tex]-axis, we set [tex]\( h(x) = 0 \)[/tex]:
[tex]\[ (x - 3)^3 + 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ (x - 3)^3 = -1 \implies x - 3 = -1 \implies x = 2 \][/tex]
The graph crosses the [tex]\( x \)[/tex]-axis exactly once at [tex]\( (2, 0) \)[/tex].
[tex]\[ \text{True (crosses once, not three times)} \][/tex]
6. Graph increasing from negative infinity to positive infinity:
Since [tex]\( f(x) = x^3 \)[/tex] is an increasing function, and the transformations do not change its overall behavior, [tex]\( h(x) = (x-3)^3 + 1 \)[/tex] will also increase from negative infinity to positive infinity.
[tex]\[ \text{True} \][/tex]
### Summary of True Statements:
- The domain of the function is all real numbers.
- The range of the function is all real numbers.
- The graph crosses the [tex]\( y \)[/tex]-axis once.
- The graph crosses the [tex]\( x \)[/tex]-axis once.
- The graph increases from negative infinity to positive infinity.
Thus, the true statements are:
- The domain of the function is all real numbers.
- The range of the function is all real numbers.
- The graph crosses the [tex]\( y \)[/tex]-axis once.
- The graph crosses the [tex]\( x \)[/tex]-axis once.
- The graph increases from negative infinity to positive infinity.
