Question

question 1

what is the range of the function (f(x) = -3^x + 5)?

(y > -5)
(y > 5)
(y < 5)
(y < -5)

question 2

what is the domain of all exponential functions?

(x > 0)
(x > 1)
(x < 0)
all real numbers

Explanation:

Analyze the range of the first function

We are given the function \(f(x) = -3^x + 5\). Let's analyze its behavior step-by-step:

1. The basic exponential term is \(3^x\). For all real numbers \(x\), the value of \(3^x\) is strictly positive:

$$3^x > 0$$

1. Multiplying this term by \(-1\) reverses the inequality:

$$-3^x < 0$$

1. Adding \(5\) to both sides of the inequality gives:

$$-3^x + 5 < 5$$

Therefore, the range of the function \(f(x) = -3^x + 5\) is \(y < 5\).

Analyze the domain of exponential functions

An exponential function in its general form is written as \(g(x) = a \cdot b^x + c\), where \(b > 0\) and \(b
eq 1\).

1. The exponent \(x\) can be any real number (positive, negative, or zero) because we can raise any positive base \(b\) to any real power.
2. There are no division-by-zero operations or even-degree roots of negative numbers to restrict the input values.

Therefore, the domain of all basic exponential functions is all real numbers.

Answer:

Question 1

• (A) \(y > -5\)
• (B) \(y > 5\)
• (C) \(y < 5\) (Correct answer)
• (D) \(y < -5\)

Question 2

• (A) \(x > 0\)
• (B) \(x > 1\)
• (C) \(x < 0\)
• (D) All Real Numbers (Correct answer)