State the domain and range for each:

f(x) = 4x + 1 and g(x) = x^2
f(x) = sin x and g(x) = x^2 - x + 1
f(x) = 10^x and g(x) = log x

To find the domain and range of a function, we need to understand the definition of domain and range.

The domain of a function is the set of all real numbers x for which the function is defined.

The range of a function is the set of all possible values that the function can take.

Let's find the domain and range for each given function:

1. f(x) = 4x + 1 and g(x) = x^2:

a. The domain of f(x) = 4x + 1 is all real numbers because there is no restriction on the variable x.

b. The range of f(x) = 4x + 1 is also all real numbers because any real number input for x will give a unique real number output.

c. The domain of g(x) = x^2 is all real numbers because there is no restriction on the variable x.

d. The range of g(x) = x^2 is y ≥ 0 since the square of any real number is always non-negative.

2. f(x) = sin x and g(x) = x^2 - x + 1:

a. The domain of f(x) = sin x is all real numbers because there is no restriction on the variable x.

b. The range of f(x) = sin x is -1 ≤ y ≤ 1 since the sine function oscillates between -1 and 1.

c. The domain of g(x) = x^2 - x + 1 is all real numbers because there is no restriction on the variable x.

d. The range of g(x) = x^2 - x + 1 is y ≥ 3/4 since the quadratic function opens upward and the vertex occurs at the point (1/2, 3/4), which is the minimum point of the parabola.

3. f(x) = 10^x and g(x) = log x:

a. The domain of f(x) = 10^x is all real numbers because there is no restriction on the variable x.

b. The range of f(x) = 10^x is y > 0 since any positive real number raised to any real power is always positive.

c. The domain of g(x) = log x is x > 0 because the logarithm function is only defined for positive numbers.

d. The range of g(x) = log x is all real numbers because the logarithm function can take any real number as input and produce a real number output.

In summary:

- For f(x) = 4x + 1 and g(x) = x^2:
- Domain: All real numbers
- Range: All real numbers for f(x) and y ≥ 0 for g(x)

- For f(x) = sin x and g(x) = x^2 - x + 1:
- Domain: All real numbers
- Range: -1 ≤ y ≤ 1 for f(x) and y ≥ 3/4 for g(x)

- For f(x) = 10^x and g(x) = log x:
- Domain: All real numbers for f(x) and x > 0 for g(x)
- Range: y > 0 for f(x) and all real numbers for g(x)