Question

function ( f ) is defined by ( f(x) = 1.5^x ). function ( g ) is defined by ( g(x) = 500x^2 + 345x ).

1. which function is quadratic? which one is exponential?
2. the values of which function will eventually be greater for larger and larger values of ( x )?

Explanation:

1. Identifying Quadratic and Exponential Functions

Step 1: Recall Definitions

• A quadratic function has the form \( y = ax^2 + bx + c \) (where \( a

eq 0 \)), with the highest power of \( x \) being 2.

• An exponential function has the form \( y = ab^x \) (where \( a > 0, b > 0, b

eq 1 \)), with the variable in the exponent.

Step 2: Analyze \( f(x) \) and \( g(x) \)

• For \( f(x) = 1.5^x \): The variable \( x \) is in the exponent (base \( 1.5 > 0, 1.5

eq 1 \)), so \( f(x) \) is exponential.

• For \( g(x) = 500x^2 + 345x \): The highest power of \( x \) is 2 (in \( 500x^2 \)), so \( g(x) \) is quadratic.

2. Determining Which Function Grows Larger for Large \( x \)

Step 1: Recall Growth Rates

• Quadratic functions (degree 2) grow polynomially.
• Exponential functions (with base \( b > 1 \)) grow faster than any polynomial function (e.g., quadratic, cubic, etc.) for sufficiently large \( x \).

Step 2: Compare \( f(x) \) and \( g(x) \)

For large \( x \), exponential functions (like \( f(x) = 1.5^x \)) outpace quadratic functions (like \( g(x) = 500x^2 + 345x \)). This is because the exponential’s growth is driven by repeated multiplication of the base, while the quadratic’s growth is driven by squaring \( x \), which becomes negligible compared to exponential growth.

Final Answers:

1. Quadratic: \( \boldsymbol{g(x) = 500x^2 + 345x} \); Exponential: \( \boldsymbol{f(x) = 1.5^x} \).
2. For larger \( x \), \( \boldsymbol{f(x) = 1.5^x} \) (the exponential function) will be greater.

Summary:

• Quadratic: \( g(x) \) (highest power of \( x \) is 2).
• Exponential: \( f(x) \) (variable in the exponent).
• For large \( x \), exponential functions grow faster than quadratic functions, so \( f(x) \) will dominate.

Answer:

1. Identifying Quadratic and Exponential Functions

Step 1: Recall Definitions

• A quadratic function has the form \( y = ax^2 + bx + c \) (where \( a

eq 0 \)), with the highest power of \( x \) being 2.

• An exponential function has the form \( y = ab^x \) (where \( a > 0, b > 0, b

eq 1 \)), with the variable in the exponent.

Step 2: Analyze \( f(x) \) and \( g(x) \)

• For \( f(x) = 1.5^x \): The variable \( x \) is in the exponent (base \( 1.5 > 0, 1.5

eq 1 \)), so \( f(x) \) is exponential.

• For \( g(x) = 500x^2 + 345x \): The highest power of \( x \) is 2 (in \( 500x^2 \)), so \( g(x) \) is quadratic.

2. Determining Which Function Grows Larger for Large \( x \)

Step 1: Recall Growth Rates

• Quadratic functions (degree 2) grow polynomially.
• Exponential functions (with base \( b > 1 \)) grow faster than any polynomial function (e.g., quadratic, cubic, etc.) for sufficiently large \( x \).

Step 2: Compare \( f(x) \) and \( g(x) \)

For large \( x \), exponential functions (like \( f(x) = 1.5^x \)) outpace quadratic functions (like \( g(x) = 500x^2 + 345x \)). This is because the exponential’s growth is driven by repeated multiplication of the base, while the quadratic’s growth is driven by squaring \( x \), which becomes negligible compared to exponential growth.

Final Answers:

1. Quadratic: \( \boldsymbol{g(x) = 500x^2 + 345x} \); Exponential: \( \boldsymbol{f(x) = 1.5^x} \).
2. For larger \( x \), \( \boldsymbol{f(x) = 1.5^x} \) (the exponential function) will be greater.

Summary:

• Quadratic: \( g(x) \) (highest power of \( x \) is 2).
• Exponential: \( f(x) \) (variable in the exponent).
• For large \( x \), exponential functions grow faster than quadratic functions, so \( f(x) \) will dominate.